I assume that the Night's Watch does not pay the Black Brothers any wages in coin for their service - they get their provisions for free, after all.
What I was wondering...how do the Brothers pay the whores in Moletown? Since they're not using coin, do they pay in naturalia filched from Night Watch' stores?
I guess some women so far north would choose such a life (given that life is relatively harsher than farther south), even if they are not paid in coin...
A lot of the Mole's Town transactions are paid by barter, certaintly, but there is coin at the Wall... not much, though, especially these days... (see following answer). Some coin comes north with the highborn brothers... someone like Ser Waymar Royce undoubtedly arrived well heeled, and I imagine families send gifts and such as well... and there's trade that goes in and out of Eastwatch...
Second; do the Night's Watch receive funding and resources from Winterfell, the crown, or both?
Some from both, certainly... but traditionally the main support of the Watch has come from the Gift, a broad belt of land immediately south of the Wall, which the Watch owns. There is more about this in ASOS. The northernmost half of this was "Brandon's Gift," the southern half "the New Gift." Historically the Watch farmed the former (the stewards) and taxed the latter.
Of course, the decline in the size of the NW and the depopulation of the Gift have both have huge impacts... again, there's stuff about this is SOS.
Third; How are the would-be septas and septons of the Faith trained for their calling? Is there some academy/religious center they can go to (perhaps the Great Sept in KL), or are they trained by local septas and septons?
Both, I imagine. Some local septons are not very well educated (like priestsin medieval Europe), but there are great centers of religious training, and the Great Sept of Baelor would certainly be preeminent among them.
Fourth; where (vis-a-vis Westeros) is the Port of Ibben?
The Port of Ibben is on Ibben, a large island nation in the Shivering Sea, the polar sea that lies north of the big continent where you find the Free Cities, the Dothraki Sea, Qarth, the fleshmarts of Slaver's Bay, etc. Yes, I will do a map one day... but not in SOS. If you visualize Westeros as a big Britain and the eastern continent as mainland Europe, Ibben is kind of up where Finland would be... except there's no Scandinavia, nothing north of the Baltic except ocean.