Sounds like you're well on the way OG.
Yes, caulk is meant to be painted over. As I said above though. it can be tricky to "lose" it enough to paint over. Quite often, you can fill, smooth maybe with the sponge, prime it with a thin "mist" coat, then give it a good coat, let it dry and see what it looks like.
If still visible, then try rubbing down.
Painting over repairs is often difficult. I would be interested to see if others on here might offer their suggestions. Some of them do a lot more decorating than I do.
As for the exterior... same thing, but use "exterior" caulk. As with interior, there are loads of good brands. You can't go far wrong whichever you buy. They all do the same thing.
When you come to re-paint the external crack, it often helps to add some fine sand to the first coat. That really does help obliterate the repair.