Welcome to the forum. Thanks for your question, and for your kind comments about the forum.
With condom-protected sex, you can be very confident you did not catch chlamydia or gonorrhea. On that basis alone, you should not be concerned about having infected your regular partner. And the azithromycin would for sure have cured chlamydia and would be over 90% reliable against gonorrhea.
Women frequently get genital irritation after sex. I'm not sure if by "burning" you mean painful urination; if so, conceivably she has a urinary tract infection. UTIs can be triggered by sex, which can "massage" vaginal area bacteria into the urinary tract, but the causative bacteria are not transmitted from the partner. On the other hand, that's not a very good explanation for only brief burning, say just once or twice soon after sex. That sounds more like physical irritation from sex. If her symptoms continue in this manner, she should see her regular reproductive health provider.
To your two numbered questions:
1) Yes gonorrhea and maybe chlamydia could be transmitted 36 hours after catching it. But for the reasons above, it is very unilkely you acquired either of these from the exposure you describe.
2) No STD is a likely cause of "burning" that occurs only after sex, or in such a random fashion. I strongly doubt her symptoms are related in any way to your recent commercial sex exposure.
I hope these comments are helpful. Best wishes-- HHH, MD
The there is no painful urination. Just irritation during sex - sometimes.
Thanks for the clarification. This makes a UTI less likely, but could indicate a yeast infection. If it continues and is new, she should get it checked out. But I'm still confident she has no STD, at least none that you picked up during your commercial encounter a few weeks earlier.