We just bought a house where the previous owner installed TRAVERTINE in the shower stall and bathroom floor.?

Travertine is a very porous, soft material, at least compared to other tiles, like ceramic. Usually the manufacturers of travertine fill the surface don't know what the filler is, but it is only for surface you see. So the installer should not only spread mortar on the floor, but also back butter the tile, so there is full contact.

I can't tell you about the tile being filled. It may be an inferior product, but the installer shouldn't have to use any fillers. It should be sealed however. It is really easy, get a bottle to mist the tile. Let it sit. Other than that, just sweep and mop for maintenance. Check with the sealer product, it will suggest how often you need to reseal. And if sealed and cleaned in a properly ventilated bathroom, it shouldn't mold, but that would be the same as any other flooring.

The paint stains are more difficult. It makes me think it wasn't sealed properly. As I said, it is porous, and will absorb stains, thus the need to seal. If you can't get the stain up will normal methods of soap, water and elbow grease, then it is probably absorbed into the tile. You can use some very fine, 300 grit or finer, wet sandpaper to see if it is absorbed deeply. I'd try it in a very hidden spot. Then you can make the judgment call if it is worth replacing. It the paint is deeply absorbed it won't come out, sorry, and will need to replace it or live with it.

Even if you think it has been sealed, I'd seal it again. That way you know the time it should be resealed, instead of guessing. It doesn't cost much and doesn't take a long time.