If i take hiv ag/ab test a day after mrna covid-19 vaccination, would there be any cross reactivity (false positivity) of the test. should i relax and take the test or wait a bit more? also, is hiv ag/ab test at 5 months post exposure conclusive?

There should not be any cross reactivity between the two tests. There is no such thing as a perfect test. They all have varying degrees of sensitivity and specificity. So your negative HIV test is really good news and would be considered clinically conclusive but there is always a very small chance that it's not.

There are no medical conditions, drugs, or treatments known to interfere with reliability of the the HIV AgAb blood tests, including the COVID vaccines. Your negative result at 5 months is 100% conclusive.