It’s from the archaic sense of the preposition with meaning “on condition that, if, with the intention that” and the verb draw, meaning to move toward oneself. So it adds an element of agency and intentionality to the movement. The original senses of withdraw, both dating from the early thirteenth century, are “to take back something that had been given” as well as “retrograde movement, to remove something from its position.”
I don’t think so. The sense of “with” in withdraw is its “original” sense of opposition ("apart" or “against"), reflecting its derivation from the Indo-European root wi- meaning “apart” or “asunder” which is also the basis of the various words identified by Dutchtoo (German wider, Dutch weder, etc.). Thus to withdraw is to “draw apart” (or “away” or “back"), while to withhold is to “hold apart”.
From the notion of “being apart or distant” rose that of “against” (as in German wider) found in withstand, literally “to stand against”. The imported word widdershins ("in a contrary or counterclockwise direction") is literally “against direction”; the OED prefers the form withershins.
See the OED entries (free access this month, as pointed out in the blog) for with- (prefix) and with (preposition), as well as that for the obsolete prefix wither- found in lots of Old English compounds like wither-crist ("against Christ”, i.e., “Antichrist").