Mammograms normally start at age 40 or 50. Yes cancer is hereditary. You can start self breast exams now.
DO NOT be afraid of pap-smears! Get a female gyno, and everything will be fine! Be sure to tell them you're anxious about it! It's nothing to be frightened of.
You probably don't have a "whole" hymen right now, just so you know. You probably broke it during sports, bike riding, using tampons or masturbating, etc. Virginity has nothing to do with your hymen. There is NO reason to feel "dirty" without one. Also, a pap smear will probably NOT break your hymen unless it is too thick to do the examination, in which case you'd want it broken before sex, wouldn't you? That would be best, otherwise your first time is going to be incredibly unhappy, painful, and not the beautiful experience it should be. I was born without a normal hymen! I never felt "dirty"