Is it normal for me to be spotting when I already had my period this month?

There's no way we could know. If you are trying to figure out if you're pregnant, wait and see if you miss your next period. If you do, wait an additional two weeks, then take a pregnancy test with first morning urine.

Some women just spot, others spot as a sign of pregnancy,

So "could you"? If you had sex, yes you could.

You have to wait to see if you miss a period or not