My last period start day was around July 22 had sex August 22 & 23 still no period. Can I be pregnant?

Possibly. While pregnancy is possible it is as likely that your periods are becoming more irregular as you get older. Less common causes of irregular periods are also possible. You can reliably do a pregnancy test in another week to detect a pregnancy from August 22, and then the general rule of thumb is to get checked if you have no period for 3 months.
Unlikely. Given the timing, but take a HPT if your period is late. If negative, you may just have a skipped or late period which is common for women once in a while. Take care!