I have uncovered some further material relevant to the first and second posted questions, although not to the third.
An early source linking 'saunter' to 'Sancta Terra'
With regard to the earliest source in English to make the connection between saunter and sainte-terre, the following entry for "to Saunter about" appears in Guy Miege, The Great French Dictionary (1688):
to Saunter about, to go idling up and down, batre le Pavé, courir les Rues, comme une personne qui n’a rien à faire [beating the Pavement, running the Streets, like a person who has nothing to do].
This Expression is thought to come from Sancta Terra, the Latine Name for the Holy Land; those being said at first to have Sauntered about, or up and down, who pretended to have been at the Holy Land.
Miege doesn't site a source for this claimed derivation, but the 1699 slang dictionary cited in TaliesinMerlin's answer makes essentially the same claim, which suggests either that some earlier print source had made the assertion or that by 1699 the origin theory was fairly well established in folk etymology. Of course, the Canting Crew slang dictionary might have relied on Miege as its source, but it doesn't seem especially beholden to formal dictionaries for most of its information.
Early instances of 'saunter' in the sense of 'wander or go idling'
With regard to the earliest use in English of saunter (or its variants) in the sense of "to wander or go idling up and down," I found several instances from before 1671 (the year of publication of the earliest instance that the OED cites).
From "Against the Arians, Hipocrites, and Disceuers," in Epigrams and Sentences Spirituall in Vers, of Gregori Nazanzen, an Auncient & Famous Bishop in the Greke Churche Englished by Tho. Drant. (1568):
Now Lordes you set aloaft, / With Pantackles, and moilles / Your fete are feanced softe, / To pageanes, and to plaies, / Ye sawnter vp, and downe, / On Theaters ye shine, / The freshest in the towne / Pretensed outward shape / Ye conterféete, and steale / But in Religion true, / Pure inwardly to deale, / You are from that as farre, / As miserable stockes, / As most infected shéepe / Of all your scabye flockes.
The context of this very early instance suggests that to "sawnter up and down" was to walk in an idle and unhurried but conspicuous manner that drew the attention of others.
From From William Wycherley, Hero and Leander in Burlesque (1669):
For she was forc'd, although 'twas not the fashion, / To put him first in mind of consummation; / And told him that about Twelve of the Clock /
She wou'd expect him with clean Sheets & Smock, / At her own Mothers House, at sign of Tow'r: / But warn'd him he shou'd knock at the Back-door; / And that her old Nurse should sit up, and spin, / To drown the noise he'd make, and let him in. / In the mean time to th' May-pole, and the Green / She bid him go to see, and to be seen, / Or where he wou'd might saunter up and down, / And count the Signs, and fine things of the Town: / Or to the Shore might go, if nothing else / He had to do, and pick up pretty Shells. / But howsoe're himself he chose to recreate, / He should be sure to keep her Honours Secret.
Here, as in the 1568 instance, saunter refers to idle wandering of a sort that invites public notice and perhaps even notoriety.
From "A Bat, Bramble, and Cormorant," in Roger L’Estrange, Fables, of Aesop and other Eminent Mythologists : With Morals and Reflections (1669):
A Bat, a Bramble, and a Cormorant, Enter’d into Covenants with Articles, to joyn Stocks, and Trade in Partnership together. The Bat’s Adventure was Ready Money that he took up at Interest; The Brambles was in Cloaths; and the Cormorants, in Brass. They put to see, and so it fell out, that Ship and Goods were Both Lost by Stress of Weather : But the Three Merchants by Providence got safe to Land. Since the Time of this Miscarriage, the Bat never Stirs abroad till Night for fear of his Creditors. The Bramble lays hold of All the Cloaths he can come at in Hope to Light upon his Own again : And the Cormorant is still Sauntering by the Sea side, to see if he can find any of his Brass cast up.
And here, sauntering might be likened to "lingering" or "loitering," as in the 1630 example, with an emphasis on hanging around rather than on moving from one place to another.
A slightly later but striking usage appears in Lancelot Addison, A Modest Plea for the Clergy ; Wherein Id Briefly Considered, the Original, Antiquity, Necessity (1677):
But not to saunter away time in sounding of Puddles, it sufficeth our present purpose to take notice, That those Provinces among the Romans over which they set Procurators, Prætors and Proconsuls, were styled Cleri. And that in allusion to this acception of the word, the Charge or Portion assigned by Lot to Matthias, whither, as most conceive, he was to go preach the Gospel, was called Κληζοι Διαηονιας και ’Αποςολης, the Lot of Ministry & Apostleship.
The central meaning of "saunter away time" in this example is "to idle away time" or "to waste time"—without any hint of physical movement. This particular figurative sense of saunter is novel for its time and does not seem to have caught on.
Early instances in which 'saunter' may have a different meaning
One very early instance of santering appears in George Joye, *The exposicion of Daniel the Prophete Gathered oute of Philip Melanchton, Iohan Ecolampadius, Chonrade Pellicane [and] out of Iohan Draconite* (1545)
We exhorte therfore as many as we may for the glorie of god that thei separat themselues from this vngodly facciō both in iugement & will: as did the Maccabeis exhorte many to auoid the company of the counsels confedered with Antiochus we warne also the lerned and prudent which yet for the studie and zele of peace (as thei wolde be sene) or for a certain singulare precise morosite wolde apere to abhorre and estiewe these new facions and soden mutacions (as thei call them) being alto ware to wise and to charely circumspecte in this their slake santering leste their rasshnes (as thei pretex it) shuld confirme the enimies of the gospell therfore decree thei thus to stād still lyke idle idols and in securite as it were afarre of loking vpon and beholdinge the bront of the bataill no handis putting forthe nor yet once (whē thei might) to helpe to any amendement or reformacion. But the mater is to manifest and to farre gone as may easely be perceiued of men of clere iugement not being corrupt with any affeccions.
The sleek sauntering in this example seems to have a lot in common (as a matter of attribute attitude) with the sauntering of the "Lordes" in Nanzan's 1568 polemic against Arians, hypocrites, and deceivers. There is an element of the flaunting fashionplate in both descriptions—but I am not altogether sure whether "santering" in Joye's case entails actual promenading or whether the author has some other idea in mind.
Even less clear is the sense of saunters in this example from William Turner, The Ffirst and Seconde Partes of the Herbal of William Turner Doctor in Phisick (1568):
Although that all they that haue writen of herbes / haue affyrmed and holden / that the Brake [fern] hath nether sede / nor frute: yet haue I dyuers tymes proued the contrarye / whiche thinge I will testefye here in this place / for there sakes that be studentes in the knowledge of herbes / I haue foure yeres together one after an other vpon the vigill of saynt Iohn the Baptiste (whiche we call in Englishe mydsomer euen) soughte for this sede of Brakes vpon the nyghte / & in dede I fownde it earlye in the mornynge before the daye brake / the sede was small blacke and lyke vnto poppye. I gatherid it after this maner: I laide shetes and mollen leaues vnderneth the brakes whiche receyued the sede / that was by shakynge and beatynge broughte oute of the branches and leaues. Manye brakes in some places had no sede at all / but in other pla∣ces agayne: a man shall fynde sede in euerye brake / so that a man maye gather a hundred oute of one brake alone / but I went aboute this busynes / all figures / coniurynges / saunters / charmes / wytchcrafte / and sorseryes sett a syde / takynge wyth me two or three honest men to bere me cōpanye / when I soughte this seede / all the villages aboute / did shyue wyth bonfyers that the people made there / & sometyme when I soughte the sede / I fownde it / and sometyme I fownde it not.
I have no idea what saunters means here, although in context one might suppose that it refers to some sort of amulet or invocation with supernatural powers.
Earliest of all instances of santer that I found in a search of Early Enlish Books Online is this one from The myracles of oure blessyd lady (1496):
And now go home vnto thy monaesterye / for whan thou comest thyder thou shalt be chosen abbot. And whan thou art abbot / teche the couent to saye my sawter / as I haue tolde the. And preche it ouer to all the people to do the same / that well it may be to them bothe in lyfe and in dethe. Seuen yere thou shalt lyue to preche and teche the people that they maye exarcyse and vse to saye my sawter. And at the seuenste yeres end yu shalt dye & come to me. Also knowe thou for certayne that many the whiche sholde haue perysshed and be loste / shall be saued by my sawter saynge. And whan she had this sayd she assended vp to heuen / and anone after this monke was made abbot. And thanne he taught his brethren &. ye people there how they sholde say our ladyes santer. & also by ye space of that-seuen yere he preched dylygentely to all folke / to kepe & vse the same forme & saynge / as he. was warned before of our blessyd lady· And whan this was done he blessedly dyed & went to heuen.
But this instance of santer seems most likely to be a misreading of sawter—that is, psalter—which appears (three times) earlier in the extract.
Instances in print of an explicit claimed connection between saunter and sainte-terre (or Sancta Terra) go back to at least 1688.
Published instances in which saunter seems to carry the meaning "to wander or go idling up and down" appear at least as early as 1568 and perhaps as early as 1545.