I went and checked out what I seemed to recall about "thee" or "thou" versus "you," and found that my memory wasn't shot. Originally, "thee" was considered a more intimate form of the word, and "you" more formal. Using "thee" in speaking to Feanor would have indicated the wish for a more personal and less formal interaction. The difference between "would" and "will" is more subtle. In this case, they are both being used in a request, but "would" implies a conditional situation (as in "I would if I could"). "Will" does not necessarily mean the question is a command, but it does not have the conditional implications of "would"; it is more straightforward ("will you or won't you"). "Will" could indicate a command, as in "you WILL do this," but Tulkas does use it in a question. I think the difference is that Manwe, because of his nature, is making a gentler (if fervent) request of Feanor; Tulkas, because of HIS nature, is pressing the issue, attempting to get a straight yes or no from Feanor. If that is the case, then their uses of would/wouldst and will/wilt (the latter being simply more archaic forms of the same word) are appropriate. Manwe is being polite. Tulkas is being a bit pushier.
And I would try to translate this, if I remembered more than a few words and phrases of Russian (studied it too long ago, alas
).