Some women are born without hymens and some tear for various nonsexual reasons, like bike riding or stretching. It is highly unlikely that a woman would get pregnant while menstruating, because the egg that would have welcomed sperm is no more. In regards to your second question, a woman can have sex as much as she wants during her menstruation cycle. There is no valid medical proof saying that a woman puts herself in danger when she is menstruating and has sex. In fact, many women who get period-related pain claim orgasms help ease their cramps.
Some women don' t like to have sex while menstruating, others do.
On the other hand some partners of menstruating women don't mind a bloody mess, others do. What matters is the comfortability of both partners when this subject arises. Many women use tampons and let their partners perform oral sex. Others do the same plus use a dental dam or type of other barrier. Others use a male or female condom.