I had unprotected sex last month then the day after I had my period. Then this month I missed my period. Could I be pregnant?

Its possible. But how old are you? If you are still young, it could simply be irregularity.

In fact, I often start my period, the very day after sex, even if my period isn't due yet. Its like a 'jump start' if you will.

You will have to wait it out and see if you miss another period, and then take a test. Don't stress just yet. But make sure you stay safe and always use protection!

Yes that is very possible, if you had sex the day before the period it is possible to be pregnant,
If you have missed this month, wait a week after and try a pregnancy test with the first urine of the morning