[BZZT!]

No Balrogs ever fell to their deaths. Two balrogs, mortally wounded, FELL. Big difference.

And note that two winged dragons died after falling from the sky, because they, like the balrogs, were mortally wounded.

: ...And why didn't the Balrog of Moria use his wings when he
: fell from the bridge and glide to a landing next to the
: underground lake instead of plunging right into it?...

For the same reason Gandalf didn't use his parachute-like cloak: the author didn't write it that way (and we don't know that it was a lake -- in fact, we have better reason to guess it was a river).

: ...Also, the fact that the battle between Gandalf and the
: Balrog continued on the way down implies that they were both
: falling at the same rate. If the Balrog had wings, why not
: slow down and let the wizard fall past him?

You're disregarding the fact that the Balrog had lashed at out Gandalf with its whip so as to drag him after it, and that Gandalf fought with the Balrog on the way down. Clearly battle, not sending him to his death, was what it had on its mind. And since Gandalf said they fell for a long time, it's a pretty good bet they didn't fall at a fast rate.

(This all sounds so familiar....)

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The Lord of the Rings and Balrog Wings, Oh My!