Old tile would need to be removed to replace. You can paint the tile with Rustoleum or Homax tub and tile paint kits.
You can paint it. Just make sure you use a paint that will work on tile
Before you paint or cover it, be sure there is no mold or mildew that you're covering just to hide it. If you don't, it unfortunately will return. A real bummer after spending time and money to cover it up
Painting tile is a great idea although I have never done it so I can't give you first hand info on how well it keeps up. You can also use a service called Bath Fitter. They can come out and place a surround and will cover all the old tile and the tub itself as well if need be. Google the company.