Definitely do not register as an escort with HMRC if you are/training to become a medical professional, you can be struck off for 'bringing the profession into disrepute'.
I'd say it's more the actual doing the occupation, rather than what you tell HMRC that's going to give you that result - which is what the OP needs to think carefully about in my above post.
Yes, but by telling HMRC you're an escort there's an increased chance of your employer finding out too.
As long as you are unrecognisable by how you blur or crop your face, and are careful not to disclose what you do to anyone you should be fine. I also make sure my photos could be passed off as nice underwear shots rather than obvious escorty ones, ie fingers in pants, topless etc. then if worst came to worst it'd be an easier hole to dig out of.
As far as I'm aware HMRC shouldn't be telling employers anything more than the TAX and NI coding required for PAYE, remainder is confidential stuff.