Does an insoluble solid dissociate into ions? Is the reaction reversible? Will an equilibrium reaction form when a chemical reaction forms an insoluble solid? Why, or why not?

To a small extent.

All reactions are reversible even though it may be very small. This one is very small.
Yes, an equilibrium reaction will form. All reactions are equilibrium reactions.
Some of these answers probably are not what your prof is looking for but they are right. We usually say a reaction that forms a ppt goes to completion and we usually don't think of these as equilibrium reactions; however, they actually are. The reactions are so far to the right that, for practical purposes, there is no reverse reaction.