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. They have odds and ends of hardware, furniture, bathroom items, etc.
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.
Look inside pots or any other containers that are out on the counter! Check under the table and other furniture, and in the cabinets and the refrigerator.
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...
Absolutely not. If you're going to paint anything with a finish on it (which your cabinets almost positively do), you definitely need to rough the material up a bit if...
No, it's just not going to work. I mean, it may work temporarily, but that paint is going to start peeling off and flaking once it dries. You really have...
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.