Welcome to the STD forum.
Responding first to the opening comments, before getting to the specific questions: STDs like gonorrhea, chlamydia, etc do not travel through the body from one place to another. You cannot catch anything by oral sex that would then cause problems in the genital area. It is not possible your testicular pain or epididymitis -- if indeed you have epididymitis -- to have come from the sexual exposure you describe. In addition, asymptomatic gonorrhea probably does not lead to epididymitis, and in any case 4+ months definitely is against any connection with the oral sex event.
Second, epididymitis generally does not cause the rather mild sort of testicular discomfort you describe. However, I was not there and don't want to imply the urgent care doctor was wrong. But my guess is that s/he suspected but did not confirm epididymitis and that the decision to treat with antibiotics was more as a precaution than out of real conviction. I believe the gradual improvement while taking antibiotics was either a coincidence, i.e. the discomfort was going to clear up anyway, or a placebo effect.
To your specific questions:
1) No. It is not possible this is due to gonorrhea or any other STD from the sexual exposure described.
2) We generally don't speculate about non-STD explanations for peoples' symptoms. But even if you have epididymitis, most cases in men over 35 years old are not due to STD but are caused by non-STD urinary tract infections (although I doubt that's what you have). Another possibility might be prostatitis (inflammation of the prostate gland) or, even more likely, the chronic pelvic pain syndrome. You should ask the urologist about these possibilities. For more information, google CPPS (spell it out) and start your reading with the excellent Wikipedia article and the information from the Stanford U. dept of urology, both of which should be near the top of the google hit list.
3) Even if you have infectious epididymitis, which almost certainly you do not, missing a couple of doses would make no difference.
Bottom line: Follow through with the urology referral, but in the meantime don't worry about an STD explanation for any of this.
Regards-- HHH, MD
Thank you so much for your thoughts. I will research CPPS.