The last time I had unprotected sex (he did not finish inside me, but there is always a chance of pre-ejaculaton containing enough sperm to impregnate, I know) was around June 30. A few days later (July 3 or so), I had a normal period (on time). Now I am a few days late on my next period. Could I be pregnant even though the period immediately following the sex was normal?

If the sex was within a couple days, of period yes, it takes about 3 days for sperm to reach egg.

The chances are very very slim, but yes there is a chance.

A few days late is not "late" it is more surprising to be exactly on time. But if no period for two more weeks, you can try a pregnancy test.

It is possible of course to even miss a period now and then also