[Boromir88]

That said, if Gandalf wasn't incumbered by the mandate and allowed to manifest as the Maia Olorin, would he have been as powerful as Sauron? Powerful enough to take him on one-on-one?

Tolkien answers this in Letter 246:

Of the others only Gandalf might be expected to master him - being an emissary of the Powers and a creature of the same order, an immortal spirit taking a visible physical form.

So, Gandalf would be expected to be the only one IF it was possible. Whether he was or not is really up to you. Gandalf was referred to as the wisest of the Maiar, but he surely wasn't the most powerful.

Did the majority of the inhabitants of Middle-Earth know what Sauron was (that is to say, a fallen Maia who had been corrupted by Morgoth)? If I had to venture a guess, I'd assume that it was pretty well known by the Elves, who'd had real-time interactions with the Valar/Maiar, but what about the other inhabitants?

This may also be up to debate/opinion. A while ago, and I had to do searching for this one I did a thread on possible who knew that Gandalf was Olorin. I figured Glorfindel knew, because back in Valinor they talked a lot, and most likely Cirdan as well, for he foresaw that Gandalf would be the only Istari to stay true to his task and gave him Narya to aid him. But, you can check out this thread, and see if it's any help:
Did anyone know?

By the time of LOTR, the Valar had long stopped directly intervening in Middle-Earth affairs. That said, if Sauron had reclaimed the ring/regained all of his power, could the Valar have still personally unseated him if they'd gotten super-****ed

Probably, but Eru wouldn't let it get to that point. As he was the one that caused Gollum to slip and fall with the Ring into Mount Doom. But, I would say that if it came to that point, seeing as Eru still cared for Middle-earth they probably would have gotten involved.

Regarding Morgoth: as Melkor, he was little short of a fallen god and was unmatched by any being in Arda (save Manwe, possibly?). Over time he bled a lot of his native power into the world, weakening himself to the point where--though still immensely powerful--he was confined to a terrible form and apparently unable to repair injuries (Silms burned his hands, Fingolfin left him limping, an eagle tore some chunks from his face). Is it feasible to suggest that perhaps one of his lieutenants--Sauron in all likelihood--might have tried to overthrow him; or was he still too powerful for something like that to happen?

Another questionable one, I would say no. However, Tolkien does say that at the end of the Second Age Sauron was more powerful than Morgoth was, because of the reason you mentioned, in that Morgoth had spread out and "bled" his powers. I will have to do some searching for the quote, as I don't have it available at this moment.

Last question: was Ungoliant more powerful than Morgoth? It seemed to me like she could have done him in when he refused to cough up the Silmarils, were it not for the timely arrival of the Balrogs.

That's what it seemed like. But, I wouldn't be so hasty it what it seems. The WK was more powerful than Eowyn and Merry and yet they were able to defeat him. In Tolkien's books it's hard to set up powers, and who is more powerful than whom, as there never seems to be "levels" as in Dragon Ball Z . At any moment anything can happen.

Welcome to the forum and I hope I was of some help.