This answer addresses only the first question of the four asked above—"Was 'Whatever tickles your fancy' originally a double entendre expression?"
Early instances of 'tickle [one's] fancy'
I consider it unlikely that the phrase "tickle [one's] fancy" originated as a double entendre. One of the earliest matches for the phrase that a Google Books search finds is from The Princesse Cloria, or, The Royal Romance (1661), spoken by man named Creses:
No sooner was I come to the presence of Hercrombrotus, but according to his usual fashion of dissimulation, he received me with a most chearful countenance, (seldom putting on other to his greatest enemies) and withal discoursed with such loving and familiar language, as if that night I should have been his Bed-fellow, pretended onely out of affectionate confidence; which I must confess tickled my fancy, however it settled not my thoughts, since I knew not how to interpret all these alterations; especially in what concerned my Lord Arethusins affairs, appearing with a face so much now contrary to his own former professions:
There is no evident sexual implication here.
Four years later, from Francis Kirkman [Richard Head), The English Rogue: Continued in the Life of Meriton Latroon, and Other Extravagants, the Second Part (1665):
And because thou art as yet but a Novice in begging, and understand not the mysteries of the Canting language, to principle thee the better, thou shalt have a Doxy to be thy Companion, by whom thou maist receive fit instructions for thy purpose. And thereupon he singled me out a young Girl of about fourteen years of age, which tickled my fancy very much that I had gotten a young wanton to dally withal; but this was not all, I must presently be married unto her after their fashion by their Patrico, (who among Beggars is their Priest) which was done after this manner.
Here the setting is sufficiently heavy with sexual interest, but the context indicates that nothing licentious can yet have occurred—and in any case it is the boy, not the girl, whose fancy is tickled. In an odd way, the fact that this setting invites innuendo makes the lack of any evident double entendre here a stronger than usual point in favor of the argument that the phrase didn't have a widely recognized double meaning in 1665.
Another early instance presents an even less plausible occasion for double entendre. From William Allen, The Danger of Enthusiasm Discovered, in an Epistle to the Quakers (1674):
But doubtless the inward Pride, the over-valuing of your selves, which I have been admonishing you of, hath had a great hand in bringing you to it [a belief "That the Light within, without being taught by man or by the Scripture, is the Rule of Faith and Practice"]. And if ever you be recovered from this snare of the Devil, it must be by being brought to a sight and sence of your own injudiciousness and ignorance, and altogether groundless confidence. And if you were but well awakened out of that Spirit of Slumber, into which your intoxication hath cast you, and out of that pleasant Dream in it, which hath so tickled your fancy, you would be ashamed and confounded before God and Men, that ever you should be so strangely deluded, and prevailed upon as you have been, to trouble the World with your whimsies and fancies, and thereby to bring a scandal upon the Christian Religion, and make sport for them that have no mind seriously to consider the things of the Gospel, and to encourage Romish Agents in their design against the Reformation.
And from William Howel, An Institution of General History, Or the History of the Ecclesiastical Affairs of the World (1685):
No sooner had he [Peuda, of Mercia] got the Power into his Hands, but he improved it for the Molestation of his Neighbours, thinking it no fault at all to disturb the World, to ruine Families, overturn Kingdoms, and destroy a Multitude of innocent Souls, and all to gratify the humour of one single Man, to tickle his fancy with the pitifull thought of domineering; the true account of the 'Actions of Conquerours, (or as that little one told the greatest of them) those publick Pirates who rob with whole Armies and Fleets, whose Power makes their Murthers and Robberies lawfull, and alone distinguisheth them from those puny ones, to which the Halter is appropriate.
Early instances of 'tickle the fancy'
In the full Oxford English Dictionary (1971 edition), the phrase "tickle the fancy" appears under a particular figurative use of the verb tickle:
5. fig. To excite amusement in ; to divert ; often in the phrase to tickle the fancy. Also absol.
It seems possible that the earliest instances of "tickle [one's] fancy" cited by Etymonline are actually instances of "tickle the fancy." The OED's first example of the phrase tickle the fancy is from 1774, but Google Books finds this instance from Edward Reynolds (Bishop or Norwich), Israels Prayer in Time of Trouble, with Gods Gracious Answer Thereunto (1645):
When I heare men magnifie quaint and polite discourses in the ministry of the word, and speak against Sermons that are plaine and wholesome, I look upon it not so much as an Act of pride (though the wisdome of the flesh is very apt to scorne the simplicity of the Gospel) but indeed as an act of feare and cowardize ; because where all other externall trimmings and dresses are wanting to tickle the fancy, there the Word hath the more downright and sad operation upon the conscience, and must consequently the more startle and terrifie.
And from Richard Baxter, The Saints' Everlasting Rest (1654):
The fittest temper for a true Believer, is to be in the spirit on the Lords Day: This was St Johns temper on that day. And what can bring us to this ravishment in the Spirit, but the spiritual beholding of our ravishing glory? Surely though an outward Ordinance may delight the ear, or tickle the fancy, yet it is the views of God that must ravish the Soul. There is a great deal of difference betwixt the receiving of the Word with joy, Mat.13.20. and being in the Spirit on the Lords Day, Rev.1.10.
Likewise, from Thomas Sprat (Bishop of Rochester), The History of the Royal-Society of London, for the Improving of Natural Knowledge (1667):
In plain terms, a universal abuse [through raillery and wit] of every thing, though it may tickle the fancy never so much, is inhuman madness ; as one of the Ancients well expresses it, who calls such mirth humanis Bacchari rebus. If all things were made the subjects of such humour, all worthy designs would soon be laugh'd out of the World ; and for our present sport, our Posterity would become barbarous.
From Nathanael Culverwell, "The Schisme," in An Elegant and Learned Discourse of the Light of Nature (1669):
And unlesse the Word of God were, as the Jewes tells us of the Manna, though very fabulously, ... unlesse the Word of God had so many several relishes, agreeable to every ones liking: Even this, though Angels food shall be loathed, and nauseated, and surely this argues a carnal Spirit. Must the νομος[?] βασιλικος bow to you and the Gospel of the Kingdome become so basely serviceable, as to do homage to your lusts? must that Word which should search the Conscience, tickle the fancy, and feed a worm of curiosity that never dyes?
Also, from Walter Vaughan, "To the Reader," prefacing his translation of Michel Baudier, The History of the Administration of Cardinal Zimenes, Great Minister of State in Spain (1679):
The Book (like the pack of Ulysses, made up of toyes for Women, and Arms for a Heroe,) though checquer'd with a pleasing variety of accidents to tickle the fancy of the lightest wits, consists chiefly of Generous Examples of solid vertue, to kindle emulation in the bravest Spirits: ...
From An Historical and Political Discourse of the Laws & Government of England, the First Part (1682):
And in continuance of time, other matters also crowded into that Society [of the Marshal's Court], although sometimes under the Directory of the Constable of England as well as at other times under the Marshal ; more particularly, that power of determining matters concerning Torniament ; a sport that like a Sarcasm tickles the fancy, but wounds the heart ; and being of as little use in a Commonwealth as of benefit, therefore is laid aside ; nor need I to speak any more concerning it.
And from John Flavel, Pneumatologia, a Treatise of the Soul of Man (1685), reprinted in The Whole Works of the Reverend Mr. John Flavel (1701):
Pliny tells us, That the Mermaids [that is, the Sirens] have most enchanting, charming Voices, and frequent pleasant green Medows, but heaps of dead mens Bones are always found where they haunt: that which tickles the Fancy stabs the Soul. If the Pain (as Anacreon well observes) were before the Pleasure, no man would be tempted by it, but the Pleasure being first and sensible, and the Torment coming after, and as yet invisible, this allures so many to destruction.
It seems very possible that the earliest wording of "tickle [one's] fancy" was actually "tickle the fancy," where the verb tickle was understood figuratively to mean something like "amuse" or "divert" and where the noun fancy was understood to refer to "imagination" or "thought" or "mental activity." The fact that so many of the earliest recorded uses of the phrase in Google Books search results come from religious figures suggests that "tickle the fancy" was not understood at the outset to allude to anything beyond its obvious figurative meaning.
The sexual double entendre of "tickle [one's] fancy" might have been used in certain circles from very early days, but any such usage evidently did not taint the phrase significantly to dissuade serious-minded authors from continuing to use it. The same might be said of the phrase as used in the present day by a great many people who have no intention of being salacious.