The garage door opener still works, and
there is a click when I turn on the wall switch. My question is (1)
Was other damage to the box besides the bulb socket?,

Possible to have fried the relay that controls the light, but not
sure.
(2) Is this socket replaceable?

I recall they are standard sockets. Mounting may be a little funny,
and you would have to improvise.

(3) How easy is it to pull it out if I can find a
replacement socket? (Or do I have to remove the box from the ceiling
and take everything apart?)

Likely, and easiest to pull the box down from the ceiling. A good
lighting shop carries various replacement sockets, or maybe even your
local hardware store. Take the socket along, along with the mounting
bracket and see what you can find.

The fact that everything works still except for the light is a
positive sign.

My son replaced the control electronics on his opener some years ago,
but with a 12 year old opener, I'd consider replacing the whole mess
one of these days, depending on how much use it has gotten.