My answer addresses the first two questions asked above: "when the term prove or proof was first used for making breads," and "if proofing dough is AmEng or just a simple spelling variant."
A circuitous road to 'proof'
Prove in the baking sense appears as a separate meaning of prove in Joseph Wright, The English Dialect Dictionary, volume 4 (1904):
PROVE, ... 7. Of yeast, dough, &c.: to rise well. [Example, from Anne Baker, Glossary of Northamptonshire Words and Phrases (1854):] It is a good yeast, it proves so well.
The source of Wright's entry has somewhat more to say about the words proof and prove, as used in Northamptonshire. From Anne Baker, Glossary of Northamptonshire Words and Phrases (1854):
PROOF. Virtue, strength; applied to the provender of cattle. "The hay is bad, there is no proof in it.
PROVE. The verbal form of PROOF: a butcher is often asked by a grazier, "How did the beast prove?" i. e. did he turn out well, was he fat? In making a cake, if it rises well, "it proves well." A baker will often say "It is good yeast, it proves so well." The former sense is given by Forby; the latter I believe is peculiar to us.
Forby is Robert Forby, The Vocabulary of East-Anglia, volume 2 (1830); and as Baker warns, Forby has the "turn out well" meaning only in the sense of livestock:
PROVE, v. "How did that beast prove?" is a question often asked of the butcher by the farmer. i. e. "Did he die fat internally ? did he tallow well?"
But that particular meaning evidently goes back to at least the middle of the eighteenth century. From James Britten, Old Country and Farming Words: Gleaned from Agricultural Books (1880):
In proof. Thriving. 'Peas less in proof.'—p. 196. See Prove.
Prove. To thrive. See In proof. 'She was thick-hided, and such beasts would not prove.'—p. 266.
Britten's source for these glossary entries is Edward Lisle, "Observations in Husbandry" (1757) who uses both "prove" and "in proof" multiple times. For example, for prove, Lisle has these examples from his chapter on "Fatting of Cattle":
§. 10. I asked Mr. Clerk how soon a calf would make beef; he said, a cow-calf would make very pretty beef at three years old, but, if killed sooner, they call it beviss ; nor would an heifer prove in fat till that time, not being past growing ; for which reason steers will not be beef till four or five years old, because they will be so long growing ; therefore it is only profitable for those countries to fat steers that plough them.
§. 16. Farmer Sartain said, he had experienced, that hop-clover and broad-clover hay would not prove a bullock in fatting;—but quære, whether this may not only hold good in the great oxen of Wiltshire.—Surely small beasts, such as are in our hilly-country [in Hampshire], may do very well with those sorts of hay.
§. 29. ... A beast should not be leather-throated, that is, have his skin hang down deep under his throat ; but should have a thin neck: the former is observed never to prove so well.
§. 34. A butcher bought a heifer half fat of me to kill : he said, she would not pay for keeping, for she was thick hided, and such beasts would not prove.—I observed the hide seemed to sit loose loose, and the hair to stare more than ordinary, or look like beggars-plush.
And for "in proof," these examples from five separate chapters:
["Wheat"] §. 21. April 14th (anno 1705) I first observed the manner of the tillowing of wheat : the spring-tillows, for the most part, do arise from the foot of the root of the winter-stems or shoots, which may be two, three, or four, according as the wheat is in proof ; they arise from that foot, and, when they break out first, they may be perceived by the eye in a bud smaller than a pin's head, containing a crystalline pellucid juice ; ...
["Foddering"] §. 21. Poor cattle may be kept to their good behavior by slight inclosures ; but by experience I find, that cattle well kept, and high in proof, must have very strong bounds, else, when they rise in case, they will soon break through, especially if they want water, or take a dislike to their pasture.
["Cows and Calves"] §. 24. ... He [Mr. Clerk of Ditchley in Leicestershire] says, if one buys in, what we call, barren beasts [that is cows that have not yet had a calf], to fat, they will require, and take bull as soon as they grow a little in proof.
["Diseases in Sheep and Lambs"] §. 13. One of the chief distempers in sheep is the ref-water, of which not one in a hundred ever recovers : it is thought to come by feeding on sour grass ; if it seizes on a fat sheep it will be worth nothing but the skin, for, if you boil the flesh for the tallow, it will stink all over the house in a strange manner : this distemper is aptest to seize on those sheep and lambs that are best in proof.
[Wood"] §. 27. The [beech] bark in the hill country will not strip so soon by a month as in the vale : again, in the same wood on the hill, there will be a fortnight or longer difference between the stripping of a tree, that is in proof, and one that is not : the sap runs fastest up a tree in proof.
As these multiple instances indicate, Lisle and his contemporaries apply the term "in proof"—meaning something like "appearing vigorous, healthy, and thriving"—to both livestock and vegetation. Lisle's book was published by his son in 1757, but Lisle himself died in 1722, and he seems to have written much of the book in the first decade of the eighteenth century. Lisle's use of prove and in proof therefore represent very early instances of those terms in connection with a vigorous or thriving animal or plant.
Early instances of 'proof' and 'prove' in connection with dough fermentation
[[UPDATE: Mari-Lou A, who is responsible for the original post (above) about where "proving/proofing dough" came from, has also uncovered what are for now the earliest cited instances of 'prove'—twelve of them, in fact, in a single book—in the context of baking. Here is the first of these instances, from A. Edlin, 1805 Treatise on the Art of Bread-Making (London, 1805):
I took four ounces of fine starch, half an ounce of isinglass, two drachms of sugar, a tea-spoonful of yeast, and kneaded them up with warm water into a paste, and set it before the fire for some time to prove ; in the course of half an hour it rose as bread usually does, and when baked, was light and porous, but more resembled muffin than bread, and appeared to want nothing but a little salt to render it palatable.
Edlin also introduces another relevant term, proving oven, described as follows:
In order to comprehend the usefulness of this improvement [Edlin's recommended plan for a bakehouse], it will be necessary to state that an oven, built upon the old principle, is usually of an oval shape; the sides and bottom of brick, tiles, and lime, and arched over at top with a door in the front; and, at the upper part, an enclosed closet with an iron grating, for the tins to stand on, called the proving oven. To heat these ovens the faggots are introduced and burnt to an ash ; it is then removed, and the bottom cleaned out. This takes up considerable space of time, during which period a great deal of heat escapes. A still farther length of time is necessary for putting in the bread, and unless much more fuel is expended than is really necessary, in heating an oven upon this principle, it gets chilled before the loaves are all set in, and the bread is, therefore, liable to fall ; a circumstance that unavoidably renders it heavy.
So already in 1805, the author can speak of a type of "proving oven" as being "built upon the old principle." Regrettably, though a Google Books search turns up numerous matches for "proving oven" from the 1800s, 1900s, and 2000s, it doesn't find any earlier than Edlin. My thanks to Mari-Lou A for pointing out this extremely interesting book.]]
John White, A Treatise on the Art of Baking (Edinburgh, 1828) uses the term proof (in the sense of "dough fermentation") as though it were so familiar to his readers that he need not define it. The first of its multiple appearances in the book occurs without any fanfare:
The most common way of procuring leaven is to mix flour and warm water into a thin paste, letting it stand for a few hours, but if it is wished to hasten the operation, a little sugar, or any other sweet, may be added, and the fermentation will take place the sooner. Whenever the mixture drops, it is then what is called leaven, and is capable of working a greater quantity of flour. For this purpose, more warm water is added, twice the quantity of what was at first used to make the leaven, with as much flour as will make it into a thicker paste than at first; after this is done, cover it close up, until it rises and drops again about an inch, then a third part more of water is taken, (hot or cold, as the season may be), with two ounces of salt to each pint of water, Scots measure, and a pretty stiff dough made,—the dough must get two or three hours proof before it is wrought down, and put into the oven.
Michael Donovan, Domestic Economy (London, 1830) provides an early definition of proof as a noun referring to the substance of the dough itself at a late stage before baking; unfortunately, Donovan provides no historical background on the term:
The total quantity of flour to be used is four hundred-weight ; but only about a third of this quantity is mixed with the liquor [composed of hot water and yeast] in the trough at first. The mixture is then well worked up with the hands until it is perfectly uniform throughout, and is quite free from lumps : this is called sponge. ... The sponge is then diffused through a quantity of water, cold in summer, and scarcely warm in winter, holding three pounds and a half of salt in solution. All the rest of the flour is then to be added, and the whole is to be well worked up into an uniform stiff paste : this is called dough. When the dough is made it is covered up, and is allowed to rest an hour and a half. It again swells, and when sufficiently spongy, it is called proof, and is fit for weighing into masses or loaves. The loaves are then introduced into the oven.
These two sources suggest that prove and proof were well established by 1830—and yet a very similar description of how to make yeast bread doesn't mention either term. From "An Experienced Method of making Excellent Bread," in The Gentlemen's Magazine (March 1758):
It [the dough] requires double the kneading the common wheaten yeasty bread does, and must afterward remain covered till it be well risen ; then kneaded again very well and formed into loaves. Some, by good kneading and raising, will make half the [prescribed amount of] leaven serve.
And likewise from William Ellis, "Of making common Wheaten-Bread for a private Family in Hertfordshire," in The Country Housewife's Family Companion (1750):
The good Housewife also observes not to heat her Water too hot, knowing that if she does, it will cause the Bread to be too heavy and close. ... But the Common Baker says, That Country Women do not understand making and baking Bread in the best Manner, because they generally, on putting their Yeast, Salt, and Water to the Meal, mix all together as fast as possible, and after letting the Dough lie but little (as some of the worse Housewives do) they mould it into Loaves, and directly put it into the Oven, without giving the Dough its due Time to ferment, swell, and rise : But the good Housewife makes her Dough ready before she begins to light the Fire in the Oven, that it may have the longer Time to lie before it is moulded into Loaves.
And earlier still, from a 1704 translation of Louis Lemert, A Treatise of Foods, in General (1702):
In the fourth place, you must for some time leave it [the kneaded dough] well covered, in a place that is moderately hot, that so it may ferment enough and swell ; but if it continue too long in this Condition, the acid Salts of the Flower having time to raise themselves considerably above the other Principles, and so to be disengaged from the oily Parts that do detain them, they do afterwards make the Bread sour.
All three of these quotations describe the proving stage of the bread-making process, but none of them use the word prove or proof. This leads me to think that those terms became popular among bakers sometime between 1758 (the date of The Gentlemen's Magazine article) and 1805 (the date of A. Edlin's Treatise on The Art of Bread-Making).
Theories of 'proof' that are regrettably low on proof
The adulteration of bread (with chalk, alum, beans, whiting, and (in some cases human) bone meal or ash) was a major controversy in London in the late 1750s. For example, James Manning, The Nature of Bread, Honestly and Dishonestly Made ; and Its Effects as Prepared at Present on Healthy and Unhealthy Persons receives a fairly lengthy (and positive) review in The Critical Review (January 1758) that notes some of his tests for detecting adulterating agents used by dishonest bakers. The criticism led to regulatory legislation—"An Act for the Due Making of Bread: And to Regulate the Price and Assize Thereof; and to Punish Persons who Shall Adulterate Meal, Flour, Or Bread" (1763). But I could find no evidence at all that the use of proof and prove in baking had anything to do with the rising of the leavened dough as a sign of its unadulterated nutritiousness. The timing appears to be merely coincidental.
Another intriguing but probably false clue appears in Noah Webster, An American Dictionary of the English Language (1828), which includes this subentry under its entry for proof:
5. The proof of spirits consists in little bubbles which appear on the top of the liquor after agitation, called the bead, and hence by the French, chapelet. Hence,
6. The degree of strength of in spirit; as high proof; first proof; second, third or fourth proof.
Might the bubbling of the fermenting dough during the last stage of rising have been equated with the "little bubbles" that appear as a result of agitating liquor? Again I have not found any evidence to support that theory.
'Proof' vs. 'prove' and 'proof the dough' vs. 'prove the dough'
In my Google Books searches, the earliest two references to proof (as opposed to "prove") in connection with baking that I found were from Britain (White's 1828 Treatise on the Art of Baking, published in Edinburgh, and Donovan's 1830 Domestic Economy, published in London). So it seems likely that proof got its start in Britain, or at any rate was in use there before it became especially popular in the United States. The term doesn't appear in any of the pre-twentieth-century dictionaries of American slang that I consulted.
The earliest match in a Google Books search for the phrase "proof the dough," where proof is a verb, is from J.M. DeWitt, "Red" in The National Baker (October 1915):
So, if I get my fermentation started with a liberal amount of yeast, and stop this fermentation just as the vegetable gases begin to lose strength, which will place the dough up to its highest point, I will have a sweet carbonic gas with a good percentage of alcohol to proof the dough in the pans with. By proofing the bread in the pans to about the same point that you proof a dough for the second fermenting period in the trough for the punching, I will have a very good expansion in the oven, and you see the texture.
The National Baker was published in Philadelphia, Pennsylvania. Other early matches for "proof the dough" appear in publications from Minneapolis, Minnesota (in 1919) and from New York City (1920). The New York City example—Bread Facts, by Ward Baking Company, is especially interesting because of they way it jumps from proof to proving to proved in a section titled "Proving the Dough":
See that you are getting a quick proof. However, avoid taking your doughs under proof as this will make your doughs stand up round and will pull away from the pans when in the oven. The bread will be heavy; it will not bake well in the oven. Caution—do not over proof the dough. A dough which has had too much proof will not show much oven "kick;" the texture of the bread may be even, but it will be coarse, the color of the crumb will be off, and the crumb will show a great tendency to crumble when cut. Do not try to get volume in your loaf by proving. A properly proved loaf if it is fermented properly will show a lively spring in the oven and will give you the proper size.
By way of a snapshot test, I looked at the first ten Google Books matches for "proof the dough" and the (only) eight Google Books matches for "prove the dough" between 1999 and 2015. The results: Of the ten books that used "proof the dough," one was from India and nine were from the United States; of the eight books that used "prove the dough," two were from Australia, five were from the UK, and one was from both. That suggests a strong split in preference between U.S. and UK usage.
I think that the likeliest source of proof and prove in connection with bread making is British colloquial usage of prove and in proof during the 1700s and 1800s to refer to the healthy growth, fattening, thriving, and vigor of livestock, crops, and trees. A similar enlarging of the dough as it sits and rises before being apportioned into pans and baked may have called forth the older rural term.
But the evidence is circumstantial at best. I haven't found any sources that claim a shared origin for the farming senses of prove and in proof and the baking sense of prove and proof. The closest thing to a direct link between the two sets of words is Anne Baker's Glossary of Northamptonshire Words and Phrases (1854), which identifies first the livestock sense of the verb prove and then the baking sense of the same word.
The Compact Edition OED (1971), though it contains definitions that allude to the colloquial farming notions of livestock that proves and trees that are in proof, doesn't seem to address the baking sense of prove and proof at all. I couldn't find any baking-specific examples, anyway.
As for "proofing dough," it does appear to be a predominantly U.S. alternative form to "proving dough"—and one that has emerged in roughly the past 100 years.