Pulling up the carpet yourself is a call only you will have to make. You could get a nice respirator, gloves, and all that gear and pull it up. The process could take only minutes.
It could adversly affect you or it could not. so, I am staying clear of that question.
But on the second question, once you get the carpet, pad, sub-floor up and replaced if needed, then I would strongly recommend vinyl or lino for the bath. It is easier to clean, easier to maintain, easier to treat for any mold or stuff, and easy to lay down.