[Foolofatook]

If the Balrog were to ensnare the Ring there is no chance that he would have handed it over to Sauron. We are speculating at the level of intellegence of a Balrog but the point of the Ring was that it would take hold of its wearer. The Balrog already being evil would not have very far to fall into claiming it as his own as Frodo did. Only the Nazgul were completely immune to this fate.

As for if the Balrog could use the Ring, I would venture to guess that he could but to the same effect as if Gandalf used it. He would use it thinking for himself but in the end it would be for the purposes of Sauron. Now, could he match Sauron if he had the Ring. Perhaps if you translate the power of the Ring to the Balrog, but in the end any power gained from the Ring is Sauron's. If the Balrog were to use the Ring against Sauron it would actually make Sauron stronger.