Could any given usage of "quoth" be replaced by "quoteth" (and vice versa)? Is quoteth simply its archaic form, or is there a difference?
Quoteth is part of the early Modern conjugation of to quote:
Today we don't use thou, and the -th become an -s:
Quoth is a variant of the third person singular of to queath. While at the time that queath was still in use (it died out around the start of the Modern Period) you could have he queþ, but there was also a variant he quoð that went back to the Old English conjugation and likely survived because "he said" is such a common thing to say (ask a schoolchild who has done some Chaucer to remember a few words from it and "quod he"—the variant Chacuer used—will quite possibly come up).
Its use has been, for several centuries, restricted to deliberate archaicism, and even early Modern use is as a lingual fossil. Even in early Modern English, "he said" or "said he" would have been more usual than "he quoth" or "quoth he".
Amusingly, in uses other than third person singular it is simultaneously both very modern and archaic, as in it is used to deliberately be old-fashioned but it is incorrect (whether through ignorance or deliberate humour) in a manner not found before the middle of the 19th Century; probably because Poe's (archaic, but correct) use made it more well-known to modern English speakers than it had been.
The two words not only mean different things, but are less related than you might think, having completely different origins.