I had always assumed that no one becaming High King/Queen after Gil-galad had more to do with the waning of the Firstborn -- their fading and the general dearth of Noldor left -- than any reflection upon the eligible candidates. I don't know if Galadriel would have taken the crown, or if even then her better instincts were to reject her desire for power, but it surely would have been a grave temptation. If we go that path, it needs to set up the final test with the Ring. I suspect it was never offered, though, not to her or anyone else.
Actually, I'd rather we tie in the fall of the High Kingship of the Noldor more to the Dominion of Men than anything else. I know the published LotR text suggests that the Fourth Age is the proper beginning of the Age of Men, but most of the Third Age action is in Gondor and Arnor, too. The elves left in ME are tarrying, and in a rather holding pattern -- at least, so far as we see. Even Rivendell and Lothlorien are depicted as bubbles outside of the influences of the time, not integrated communities within the larger world.
As to when Galadriel first desired the Ring... I actually suspect the conscious temptation is rather fresh. That is, I don't think she was sitting around, in her darker moments, longing for or wishing for the Ring while it was assumed lost at the bottom of the Sea (I see no reason to believe she wasn't fooled by Saruman's words on the subject). But when word reached her that it was found, and on its way to cross paths with her... then, I think, her old temptation for power took a new and dangerous form.