With oils, I think I've read oils in burners are fine, it's on the skin where topical application is more potent that the issue may be. So I guess intensity is the issue..
Double yes on this. The amount of any "oils" or similar would be next to nothing, if not nothing. It's when you start applying specific oils directly onto your skin is when you start to have problems.

There is so much water in the bath, so go ahead! I couldn't live without mine.