Your intuition for the first one is correct; the extra charges are held to the surface by the electrostatic force, which is many orders of magnitude stronger than the gravitational force.
The second one (and part of the first one): You're confusing free electrons with "extra" electrons.
In a conductor, the highest energy electrons are not bound to any atom, and instead float around. At any point in time, they are significantly contributing to neutralizing the field of 5-8 other nuclei (which are surrounded by less electrons).
If all free electrons fell down, then the positively charged nuclei at the top would have less electrons around them, giving rise to a strong electric field and an increase in potential energy of the system (which will never happen on its own). Basically, if a free electron leaves the region it is in, another one must come and replace it by entering the general area to balance out the forces. Failure to do so will create a hole, which will move around till it gets neutralized.
The same thing goes for moving all charges to a surface. When it comes to a conductor, only the "extra" charges must be at the surface. The body of the conductor must remain electrically neutral, and this can't happen if the free electrons move to the surface.