Welcome to the STD forum.
You describe a pretty common situation: one member of a couple with an STD, the other member apparently free of disease. There are several potential reasons and some of your questions refer to them. Before answering those questions, here are the usual explantions.
a) Your gonorrhea test result is falsely positive. With modern tests, this is uncommon, but it's possible.
b) Your partner's test is falsely negative, i.e. he is infected but the test missed it.
c) You have had gonorrhea for 5 months (yes, that's possible) and by luck your partner hasn't caught it yet. I don't think this is likely; your new symptoms suggest a new infection.
d) Your partner gonorrhea (and might have infected you), but it went away before the test was done. Without treatment, gonorrhea usually goes away in a few weeks, especially in men; or maybe your partner had antibiotic treatment he isn't telling you about, or he could have had an antibiotic for something else that cured it without ever knowing the gonorrhea was there.
When gonorrhea appears in a relationship that has been going on more than 2-3 months, usually it means one partner or the other has had other partners. But not always. To your specific questions:
1) Your facts are correct. I doubt you have been infected for 5 months; see my comment above.
2) Immunity to gonorrhea is rare. See (b) and (d) above; most likely it's one of those explanations.
3) Yes; see (b). I think this is the case here, given the antibiotic history etc.
4) Augmentin is excellent gonorrhea treatment and amoxicillin is pretty good by itself. However, trimethoprim usually doesn't work. So your own explanation is a pretty good one.
5) Without pain in your lower abdomen or internal pain during sex, PID is unlikely. But if in doubt, it can't hurt to see a gynecologist.
The mystery of who was infected first, and where the infection came from, may never be solved. However, you should talk frankly (but sensitively) with your partner about whether he has had sex with anyone else in the last few weeks.
Most important, he needs treatment for gonorrhea, regardless of his negative test result. Also, you both should be tested for chlamydia and treated for it; chlamydia accompanies 20-40% of gonorrhea infections. Discuss this with your health care provider(s).
Best wishes-- HHH, MD