Is a woman more likely to transmit an std when she has unprotected sex during her period due to the blood coming out and potentially going into penis?

Not particularly: Of the dozen or more germs that can pass during intercourse all can do so in the normal fluids generated by the activity. Menstrual blood does not enter the penis to a significant degree but will coat it. It may actually slow down some of the germ action to a minor degree. Many women do have a flare of STD symptoms during or just after menses.