Can I get pregnant the third day into my period?

Period sex is normally perfectly safe if you have cycles around 26 days or longer. It is theoretically possible that your cycle can be disturbed and you can ovulate earlier and get pregnant. In practice, it does however rarely happens.

It all depends on how low his sperm count and other parameters are. As long as there are healthy motile sperm, there is a possibility that you can get pregnant. It is unlikely, and it can take a long time, but it is possible. So don't use a low sperm count as a birth control method.