Yes, you can paint over stained wood. This shouldn't come as a surprise, because wood stain is designed to be top-coated with a clear finish, and when it comes to adhesion, there isn't much difference ...
If you have to remove any paint before staining the piece, though, there will be more steps, tools (and perhaps chemicals) involved. The steps below explain how to stain wood furniture ...
If you plan to keep furniture in front of the area and like having wood paneling, you could just paint the stained area and use a faux painting technique of lightly brushing on a lighter or darker ...
Chipboard is another name for particle board, a manufactured product, consisting of wood particles held together by a binding resin. To stain and paint chipboard, you should use particular preparation ...