Should I go to doctor for bad knee pain after slip and fall?

Yes, if persists. If the pain is limiting your mobility your should have it checked out.
Probably. A good idea particularly if it is not getting better and you are having trouble bearing weight or walking. Initially try ice, nsaids, crutch or cane and an ace wrap or brace to see if it helps if it is not better in 48 hours you might want to see a doc for an exam.