His staff he merely uses to make the bridge break (and not to light the place up, which was the case in the crappies/movies).
Not quite (although I can't recall how much light there was in the film but I suppose filming in complete darkness is a bit of a problem) - Gandalf used his staff for light during the journey through Moria described as 'will-o'-the-wisp' and on the bridge of Khazad-dum
Glamdring glittered white in answer. There was a ringing clash and a stab of white fire.
The staff broke asunder and fell from his hand. A blinding sheet of white flame sprang up.
The light came from somewhere - I think from the broken staff.
I agree that the Balrog was killed by the fall after much fighting involving swords (I assume Gandalf hung onto Glamdring somehow)
and ever I hewed him
which Gandalf did when the balrog's fire was quenched, and magic
as suggested above by Aelric and Vir.
Thunder they heard, and lightning, they said, smote upon Celebdil, and leaped back broken into tongues of fire. ... A great smoke rose about us, vapour and steam. Ice fell like rain.