Oil based paint is not necessary for most interior applications. In some cases you would use an oil based primer if you needed good penetration and stain hiding characteristics. Modern latex paints...
You can. However the old surface needs to be prepared so that it is clean and free from cracks and flaking. Giving it a good sand would probably also be helpful
Stains tend to be semi-transparent, so get rid of as much of the original water based stain as you can. Remember, the more work you do in the preparation stages, the better the finished result will...