I would seriously consider replacing the cabinets before putting a new granite countertop on them. The problem is that no one will remove the countertop at a later date if you want to change the cabinets, due to the possibility of breakage. The only choice you will have then is a reface. Adding new doors, drawer fronts, and veneered sides is just a facelift. You could also add new hardwood drawers with undermount cushion close slides, but by the time you do this if will be the same cost as new cabinets now. The cabinets you have now, frameless, particle board sides are notorious for having the hinge screws strip out the particle board they are attached to. Since you can't put a longer screw in them, because the material is only 3/4" of an inch thick you have a problem. You also can't put a wider screw in them because it won't fit through the pre-drilled holes in the hinges.

I don't know the answer to your playhouse dilemma, I just wanted to comment to follow along with the brilliant answers you receive to this very interesting window question :)

When you said hide, do you mean for it to be out of sight when you want to? If so, then you should install a cover panel. Couple of ways - the conventional one is a sliding or hinge type (like the top kitchen cabinets you see now in the market) or you can choose to have a sort of translucent glass and put everything behind it. Only when you switch on the TV, then you can see it .

It looks to me as if the hinges have been exposed to moisture. On a very tiny area, try a few drops of lemon juice and some table salt. Rub gently with a cotton swab. Wipe with a damp cloth and dry immediately. I do not know it it will work because I do not know what kind of metal the hinges are.