Yes it is possible to paint formica counter top. You would have to ask at a paint store.
Painting a counter top is not very hygenic. The paint can crack and peel and bubble (especially if something hot is placed on it.
In my house, I painted the cabinets and replaced the countertop. A countertop replacement is not too expensive or difficult.
Formica countertops with backsplash run about $10 per linear foot. Of course there are very expensive counter tops also but it sounds like you want to save a few dollars.