[Keeper of Dol Guldur]

Absolutely right, were the Balrog to take such a thing as Narya, it could never wield it. As an original creature of shadow, I'd be willing to say that Balrogs, like Morgoth's other creations, weren't exactly big sunlight fans, and so since (if Gandalf's flashes of bright white light were from use of the ring) it's power would hurt it, I would say that it would have a few options.

It's first option, the obvious one for something like a Balrog which clearly doesn't have too much of a love of menial possessions, would be to destroy it. Now, by destroying it, the Balrog may have had it in its power to absorb Narya's power (feeding on light, to belch out more shadows, kind of like Ungoliant but obviously not the same). Or it could have just wrecked the thing.

No, the Balrog and Sauron weren't on the 'same team' because that would mean the Balrog was on Sauron's team, and no way would an elfbane like that kneel to a peer. Personally, I think the orcs referring to the Balrog with cries of 'Ghash' were not all that unalike to referencing Saruman as Sharkey or referring to anything back home as 'Lugburz' wanting a prize. Those orcs may have been sent to Moria by Sauron a few centuries before to investigate and inhabit the place, but it was clear that Durin's Bane was in charge, and they kept clear of him. However, if he felt the need to use them to send message back and forth to Dol Goldur, there's no evidence of it happening. While Sauron maybe of all people knew the shadow of Moria was the Balrog, the Balrog seemingly, knowing full well Sauron was a higher up, didn't care to leave Moria for any reason. It was . . . "His realm" and any orc there was there by his leave. I wonder if he had declared himself Lord of Moria . . .

So anyway, regardless of being aware of eachother I don't think there was a lot of talk between the Balrog and Sauron. After all, the Balrog would probably take Sauron for a traitor after going AWOL when Huan ripped his throat out and leaving Morgoth without his most devious schemer during the War of Wrath. If anything, Durin's Bane probably didn't really like Sauron that much anyway, thinking him nervy to take the title of 'Dark Lord'.

As for the ring, I think Gandalf may have known he could kill the Balrog. It's likely he also knew he'd probably die in the feat, but it was he who chased the Balrog through the lowest dungeons from the lake, up the endless stair, and to the tower, not the other way around. Size clearly doesn't matter when someone is packing elf-weapons of the greatest make.

Back on the subject . . . if the Balrog at least claimed the ring, not to wield but as his trophy, he would likely know the locations of Caras Galadhon and Imladris, and then Galadriel and Elrond would know that he was a Balrog, before even the Fellowship came to Lothlorien. Not knowing how the mind of a Balrog works other than wanting to destroy opponents and serve Melkor, I have no idea what he'd do with that kind of information. Probably tell the orcs where to go . . . Maybe lead the attack on Earendil's son's domain himself. Could the waters of Hithaeglir quench the flame of Udun?