Yes it should be okay, depending on what is in the cleanser. Make sure the cleasner does not have any solvents that could cause de-lamination or damage.
If you have a Habitat For Humanity in your area, they often have stores that sell used items to repair your house. We bought several of our kitchen cabinets there....
I mean, you can do whatever you'd like, and there are certainly a lot of people out there who do this. But you're right to be a little suspicious about...
Of course! This would require a wood primer to create a smooth finish, though. You can sand the surface smooth, apply an oil-based primer and then an oil-based paint.
