You can get pregnant while on your period. Your period doesnt mean that you are free to have unprotected sex. So be careful. I'm not sure about your case with it stopping after a day, but maybe you should contact a healthcare professional about it. Good Luck, and I hope I helped!
If you are taking the pill every day at around the same time you should be fine regardless if you are on the sugar pills or not. Many women who have sex when their period is due bring it on sooner with the intercourse. Don't worry, your chances are very slim since you did start the period which is an indication of not being pregnant.