In The Fellowship of the Ring film, why did Gandalf willingly drop down after the Balrog? He had a really firm grip on the rocks, and it looked like he could easily hold on.
He didn't willingly drop down in the books:
With a terrible cry the Balrog fell forward, and its shadow plunged down and vanished. But even as it fell it swung its whip, and the thongs lashed and curled about the wizard's knees, dragging him to the brink. He staggered and fell, grasped vainly at the stone, and slid into the abyss. 'Fly, you fools!' he cried, and was gone.
The fires went out, and blank darkness fell. The Company stood rooted with horror staring into the pit. Even as Aragorn and Boromir came flying back, the rest of the bridge cracked and fell. With a cry Aragorn roused them.
This is the book version of course, but it clearly shows that he "vainly" grasped at the stone and was unable to hold on. He fell into the abyss unwillingly, giving the warning for the company to flee.
In the movie, he seems to have a more firm grip. However it does make the point (as happens in the book) to show how utterly exhausted Gandalf is during this scene. In addition to him frequently leaning on his staff, you can just tell that the man is exhausted. In this sense, it is possible (in the movie) that he decided to let go so as to save the Company from running to his aid only to be shot at by orcs, as he knew he did not have the energy to hold on that long.
If you remember the scene in the movie Aragorn barely skips up the stairs past several arrows from across the abyss, had they gone to Gandalf's aid they would have been closer and under less cover from the archers.
Gandalf is one of the Maiar...and all of them are charged with overseeing Middle Earth as a safeguard against the power of Melkor. After Melkor's death in the First Age, his agents were still scattered across Middle Earth unaccounted for. That includes Sauron, orcs, Balrogs, Dragons, goblins, etc. Gandalf and the other wizards were there as balance between those forces and the peaceful forces (hobbits, elves, men, dwarves, animals, etc.) of Middle Earth. That being said, Gandalf HAD to confront the Balrog because it was naturally drawn to him; they shared the same level of existence as a Maiar. Even if the Fellowship did not run across the Balrog, the wizards eventually would have to delve after it to get rid of it outright and keep it from being a threat to middle earth. It was the Dwarves that freed the Balrog in the first place. The wizards knew it was there, it just wasn't on the list of priorities at the time; after all-they already were investigating the possibility of Sauron manifesting himself.
So, after Gandalf fell he could have just tried to run away and escape. The problem is one, the Balrog was gonna be drawn to him anyway as a natural enemy and two, he and the other wizards would have to come back and defeat it anyway at a later time. The risk of defeating it at a later time is that the Balrog could have gotten out and caused more destruction, Sauron could have recruited it and used it in the war, or it could have disappeared entirely and caused random destruction all over Middle Earth until they could find it again. The Balrog was not the brutish creature as portrayed in the movie. It had knowledge of itself and its purpose. It recognized Gandalf and his power. It had knowledge of weapons and their uses. This denotes intelligence.
With that being said, Gandalf could NOT take a chance of it getting loose from Moria. That's why he fought it on the way down. Fighting it on the way down was a strategic move; the Balrog couldn't keep up with Gandalf while falling. Gandalf's initial plan though, was to stop it at the bridge; this was because at the time Frodo's mission with the ring was far too important. Can you imagine what could happen if the Balrog ended up with the One Ring?! That's a chance Gandalf couldn't take. He wanted to drop the Balrog and continue on with the Fellowship; because he KNEW that the journey would get even more perilous as they got closer to Mount Doom (especially since Saruman had already turned on them).
I think Gandalf becoming the White Wizard was reward from the Ainur for staying faithful to his purpose while also restoring the balance of power in Middle Earth after Saruman's betrayal to his purpose. Sorry for being long winded. I love talking about these books!
Gandalf dropped because the fellowship were looking like they were coming back into the open on bridge to try to rescue him. They would have been peppered with arrows and the chance of getting to him before he fell or was hit by an arrow himself was very low.
What would have happened if Frodo had come runnning out and been hit by an arrow or other had been hit and Frodo was left to do the quest without combatants to protect him and friends to support him?
Well for many reasons. I think one of the most important, was for his transition to become Gandalf the White. Whether he did this knowingly or not ,that's for you to decide. But the way in which Gandalf operates always seems to be premeditated, predestined. Another reason, probably thought by most people, was to allow the others to escape, which they naorly did, in fact I think it is so that they could go on the journey themselves, make mistakes because that's what makes this such a great story. They all have their own demons to deal with, and if Gandalf was with them at all times, They would never be able to complete their task.