Wægn-wyrhta is found in Old English, so wainwrights have been around in the language a long time.
Wain and wagon tends to be used for those that carry things, carriage more often for those that carry people, and coach generally for those carriages that are covered. There’d be exceptions, but it would make coach and carriage a bit more respectable than wain and wagon, both for the owner, and the tradesman who makes it.
William Felton tells us much in his treatise, more than I care to read through myself, but how a tradesman became a tradesman seems to be left as an assumption. (I freely admit I may have just missed it).
Most skilled trades were taught through apprenticeship from the mediaeval period, until industrialisation changed a lot of such professions, and changes to standardisation changed a lot of the rest. The process does still remain in some cases, and has left a mark on others.
An apprentice would train under a master. He (and in this profession, it almost certainly would have been he) was mostly compensated in room, board and training, with little or no allowance beyond that.
Upon completing his apprenticeship, he became a journeyman who could charge a wage for his day’s work.
A journeyman could submit a masterpiece to be judged by the appropriate guild, which would allow him to join the guild in his own right as a master. A master would generally be an independent tradesman, paying a fee to the guild, and being responsible for the purchase and maintenance of his own tools, premises, and so on.
In London in 1630, a petition was made for an organisation of Coachmakers and Wheelwrights to be incorporated. The Worshipful Company of Wheelwrights and Worshipful Company of Coachmakers and Coach Harness Makers were eventually incorporated separately in 1670 & 1677 respectively, and gained the status of Livery Companies in the 18th Century. They exist to this day, but are now charitable organisations, whose websites are http://www.coachmakers.co.uk/ & http://www.wheelwrights.org/
(How the guilds served as trade union, standards body, mutual support society, pension fund, life assurance provider and fraternal organisation, and came to sometimes be only one of those, and sometimes replaced by them, is a topic onto itself).
A quick look finds a period as a “journeyman carriage maker” described in the life of one Hiram Mish White (1830-1903) in biographies from Franklin County, Pennsylvania.
I’m sure a more diligent look would find examples closer to the period you are interested in.
A similarly quick look finds “journeyman coach-makers” mentioned, such as in The London Tradesman: Being a Compendious View of All the Trades, Professions, Arts, Both Liberal and Mechanic, Now Practised in the Cities of London and Westminster. Calculated for the Information of Parents, and Instruction of Youth in Their Choice of Business by R. Campbell Esq. (1747) which tells us much (all emphasis his):
How long Coaches and Chariots have been in Faſhion in this iſland, I am at a loſs to find; though I am apt to conjecture we knew very little of them till after the Norman Conqueſt. Chariots of War are of an old Invetion ; the moſt antient Hiſtorians make mention of them ; but Coaches, or, as the Quaker affects to call them, theſe Leathern Conveniencies, I believe have been but a few Centuries in common Uſe in Great Britain. We have no got ſeveral Sorts, Shapes and Figures of them, and the Art of Coach-making is arrived to the utmoſt Perfection.
The Coach-Maker’s proper Buſineſs is to make the Body of the Coach, and all the Carriage except the Wheels ; his Trade is compounded of the Carpenter, Taylor, and the Shoe-Maker ; he finiſhes his Work by the Aſſiſtance of the Founder, Tire-Smith, Wheeler, Carver, and Painter : He is a Carpenter, as he frames the Body of the Carriage of Wood ; a Taylor, as he lines the Inſide with Cloth ; Silk, Velvet, or other Materials, to which he is obliged to uſe his Needle ; and he is a Shoe-Maker, as he covers the Top and Sides with Leather, in which he is ſometimes obliged to uſe his Awl. This is a Coach-Maker’s proper Buſineſs ; as to reſt of the Work, it is finiſhed by Tradeſmen who know nothing of his Art, and apply themſelves only to particular Articles.
The Coach-Maker is a genteel profitable Buſineſs both to Maſter and Journeyman ; but requires a great Stock of ready Money to ſet up and continue Trade ; they deal with none but Nobility and Quality, and according to their Mode muſt truſt a long Time, and ſometimes may happen never to be paid. I cannot apprehend that it requires any notable Genius to form a Coach-Maker, ordinary Talents will do the Buſineſs ; it requires Strength, and a Youth can be of little Service to himſelf or Maſter till he has arrived at the Age of Fifteen ; unleſs he is of a more than ordinary robuſt Make. The Wages of a Journeyman Coach-Maker, if good for anything, is a Crown a Day ; nor is the Trade over-and-above ſtocked with good Hands.
(A crown was 5/- so that would be £0.25 in modern British currency).
This goes some way to answering a lot of your questions, though I note that coach-maker seems to be more often the British, and carriage-maker more often the American. Ngrams seems to bear this out, but I wouldn’t have full confidence in that. It could perhaps reflect a greater percentage of covered coaches among the carriages used in London, a difference in the American and British English of the time, or a combination of both. Or it could just be chance leading me to find more American examples of one, more British of the other.
Campbell’s book goes on to mention wheel-wrights (a "Coach-Wheelers" and separately talking about "Cart-Wheelers" whose work need not be as well-finished), "Coach-Carvers" and "Coach-Founders" who specialise in carving and metal-founding for coaches, and so on, while the book you yourself cites has prices for some of the items that must be purchased. Between these (and with some modernising of Campbell's 18th Century Tendency to write all Nouns in Capitals which was the Fashion among some at the Time, but already dying out), you should be able to build up a reasonable picture of the trade.