You clearly had
urethritisChlamydial urethritis - male
Urethritis, either gonorrhea, chlamydia, or non-chlamydial nongonococcal urethritis (NGU). Gonorrhea almost always causes symptoms within a few days, so it is much less likely than the others. Ideally, you should have been tested for both gonorrhea and chlamydia, but many doctors skip that step and just give treatment for all possibilities. To answer your specific questions:
1) Modern HIV tests usually become positive in 2 weeks and virtually always by 4 weeks. 3 months is more than enough time.
2) I had to look up Tavanic and found it is the trade name for levofloxacin in some countries (Levaquin in the United States). If given in a single dose of 500 mg, it would not cure chlamydia (but see no. 4 below). It usually is effective against gonorrhea, but resistant strains are increasingly common in much of the world. But if your symptoms cleared up, you can be sure that if you have gonorrhea, it was cured. Levofloxacin is unrelated to penicillin and is safe in pen-allergic persons.
3) Yes.
4) Doxycycline for a week (usual dose is 100 mg twice daily for 7 days) is highly reliable against chlamydia. (Levofloxacin also would be effective if given for 7 days, but not in single doses.)
You should do fine. "Pro" or not, you should inform your partner of 3 months ago of your diagnosis, if you know how to contact her. And of course any other partners between then and now.
Good luck-- HHH, MD