If I have a piece of my gum hanging off my tooth, should I try and cut it off or leave it be?

Sounds v strange. C a dentist find out what is going on.
See your doc. Spotting after intercourse suggests bleeding from the cervix. The most common causes of this are stds and cervical cancer. You need to be examined by a doc to know for sure what is causing this and to receive proper treatment.
See a Dentist. Do not attempt to remove this yourself. A dentist should evaluate to see the best way to treat the situation.