You'll be trapped, I'm pretty sure. But an elevator won't drop unless the cables snap, and that unlikely.
Yes, you might get trapped in it. No it cannot fall because of the safety latches and the emergency brakes it has for these events. The safety latches are held open by the weight of the car hanging on the cables. If the cables go slack, the spring loaded latches will lock the car in its present position. They are required to be inspected periodicaly.
I suspect disabling the fire alarm would be illigal.
We can try to prepare/prevent for any danger and leave the rest to god's. You have to have faith in the system you live in, like the elevator, electric, alarms, water, even a/c.
If you are so concern with building fires, or electrical troubles, consider being on a level high enough that a fire truck ladder can reach. Being on the 10 fly would be as dangerous as the second floor, but a lease you can leave by stairs.
i would hate to be trapped in one while i have to take a wicked crap , that would be a drag and really hope i'm alone in the trapped elevator then but with my luck i would just have done it and the doors would then fling open and all the building staff would be there looking at me , ha ha