Most woman don't ovulate during their periods. Its usually about 10-14 days after the first day of their period. However she could have an irregular cycle and sperm in perfect conditions can last up to 5 days. Also, birth control if taken every day at the same time is very effective however if she even misses one pill the rate of protection drops. If you don't want children at this moment in your life I would suggest not doing it. Also, I'm not sure how long you two have dated but having sex during a period is a great way to get STD's.
I would not have any form of unprotected sex at any time of the month until she has been on birth control a little longer. I know it probably says it's effective the first month (I dont know) but I would wait one more cycle just in case. And as far as having sex during her period - I agree with Alaysha. Good way to get STDs, I'd wear a condom if you're doing it that time of the month.
Anytime you have unprotected sex it can cause pregnancy. Only do it if you are ready to be a dad and be connected to this person for the rest of your life...Although women don't typically ovulate during or near their period, there is always a chance...My best friend got pregnant this way and I've seen many women on this site that got pregnant that way too. Not to mention STD's etc.