In a metal, why don’t the free electrons fall to the bottom of the metal due to gravity?
Also, charges in a conductor are supposed to reside on the surface so why don’t the free electrons all go to the surface?
Regarding the first question, my hunch is that it is due to the electric force which is greater than the gravitational force and electric force act along the length of the conductor.
I don't have any intuition for the second.
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.
You're right that the electromagnetic force is much stronger than gravity. For a particle like an electron, the electric force is around $10^{42}$ times stronger (a simple calculation shows this) than gravitational force between two electrons. The variation is simply due to the difference in both of their constants and mass & charge of electrons. Still, this doesn't mean that electrons don't curve spacetime. The effect is too negligible to measure.
Regarding your second question, a conductor doesn't necessarily have excessive charges on its surface unless the "charging" is done by you. Even if there are some static charges, it doesn't affect the free electrons significantly. Because the potential difference which drives them are stronger than the electrostatic attraction between a pair of electrons.
