Welcome to the forum. Thanks for your question.
Reacting first to the title: A negative HIV PCR test 13 days after the last possible exposure is highly reliable (probably around 99% or better), but proof will require additional testing, such as an antibody test at the 4 week mark. The combination of negative PCR at 13 days plus antibody at 4 weeks will be 100% reliable.
Now I have read the question itself. I would guess this was a pretty high risk exposure; street-working (or bar-working) female commercial sex workers in New Orleans probably have substantial rates of HIV. The risk might be especially high if she is African American.* Still, even the highest likelihood is under a 10% chance she had HIV, and most likely under 1%. So given the occurrence of unprotected vaginal sex, having an early HIV test by PCR was not a bad move. To your specific questions:
1-3) See above. It is exceedingly unlikely you caught HIV. You put yourself at moderate risk.
4,5) You don't say exactly what test you did at 31 days. If it was an antibody test, the combination of results will be 100% proof you were not infected. If it was another PCR test, you wasted your money. Given all the information, including your negative PCR at 13 days, the chance you caught HIV is no higher than 1 in a million, but you need an antibody test to nail it down for sure.
6) Your symptoms are not typical for HIV, and the negative test results prove HIV isn't the cause. (You can't have acute HIV with symptoms and still have negative PCR.) Whenever someone suggests his or her own symptoms have an emotional origin, usually s/he is correct.
7) No medicines, including flu vaccine, have any effect on HIV test resliability.
At this point, you shouldn't be at all worried. Have a final antibody test to prove it for sure. But you can definitely expect a negative result.
Best wishes-- HHH, MD
* For interested readers, the higher risk of HIV in African Americans is not a racist position, just a statement of fact; see the threads below: