Welcome to the forum. Thanks for your question.
Your initial symptoms were typical for nongonococcal urethritis (NGU, which is the same as nonspecific urethritis, or NSU), both the nature of the discharge and the timing. Are you sure about the diagnosis of gonorrhea as well? It's pretty much impossible to diagnose both gonorrhea and NGU simultaneously; if gonorrhea is found, then by definition the problem isn't NGU. My guess is the GUM clinic treated you for both but really only diagnosed NGU (unless a later gonorrhea test returned positive).
In any case, you were impressively treated with a mix of antibiotics that was obviously designed to cover every possibility, i.e. NGU, chlamydia, gonorrhea, and eventually trichomonas. It often takes several days for symptoms of NGU to start to improve, and persistent symptoms at 3 days is not unusual. My guess is you didn't actually have trichomonas and the problem would have improved without the metronidazole -- which was prescribed in an unusually large and prolonged regimen. (Usually it's a single dose of 2 grams.)
When symptoms of NGU persist after such intensive treatment, it generally does not mean any infection is actually continuing. And it appears to be harmless. Men with persistent/recurrent NGU seem not to develop any complications of any sort, and their partners don't either. In your case, it isn't clear to me that any important problem is continuing. Discharge with defecation isn't necessarily abnormal (see the link below), although you might have a case of non-infectious prostatitis. This also is generally harmless.
To your specific questions:
1) As noted above, I doubt you have any continuing infection of any importance.
2,4) Those test results are 100% reliable, showing you didn't catch HIV and I see no need for further testing. But you'll have to decide whether to follow this advice or that of your GUM clinic.
3) If you didn't have sex with your regular parnter before you completed your various antibiotics, you can safely assume you didn't infect her with anything.
5) I can't comment on a questionable diagnosis of genital warts. If you really have warts, you can expect they will soon become more obvious. And if you have warts, you didn't catch them during the sexual exposure 6 weeks ago. It takes a minimum of 2-3 months after catching HPV, and usually 6+ months, for warts to become apparent.
Bottom line: I don't think your current symptoms are worrisome. But if they continue or you remain concerned, consider seeing a urologist to check out your prostate gland.
Best wishes-- HHH, MD