The Great Evils of the Third Age

[Zigûr]

"Why did the Balrog never leave Moria?" is an interesting question. Two possibilities come to mind.

1) It didn't want to - it was contented where it was.
2) It feared what lay outside - who knew what threats it might face if it made its existence too publicly known? Dwarves were one thing, Lords of the Eldar another.

I think its presence in Moria was very convenient for Sauron. All Orcs, directly or indirectly, served Sauron by the end of the Third Age, and given that the Balrog seemingly did not object to their presence it must have been logical to take advantage of the location.

I'd argue that Sauron may not have known for sure what Durin's Bane was, although it probably wasn't especially difficult to guess. But I don't think Sauron needed it. Smaug would have been useful - I think any arrangement there would still have been more of an alliance than Smaug directly serving Sauron - but evidently he was not necessary either. Really the only thing that mattered was recovering the One Ring so that he could dismantle the defences of Lórien and Rivendell. He presumably thought that everything else could be accomplished through sheer weight of military numbers.