Welcome to the STD forum. I'll try to help.
Your doctor was correct to treat you, and azithromycin (Zithromax) is one of the recommended drugs. You really should have been treated the first time your partner developed nongonococcal urethritis (NGU).
You can find quite a bit of information about NGU on this forum; just enter that term (or spell out nongonococcal urethritis) in the search link. NGU is the single most common STD in men, by a large margin. About one third of cases are caused by chlamydia, but the specific cause in most cases is unknown. Treatment of men's sex partners is routine.
In terms of your health, most likely there is no serious risk. The mostly unknown causes of nonchlamydial NGU have not clearly been shown to carry any serious health risks for women. On the other hand, the possibility of serious consequences cannot be ruled out and treatment is recommended both to protect the partner's health and to prevent reinfection of the male partner.
To the specific questions:
His infection came from you, not anything to do with the condom. The cause was a sexually transmitted bacteria -- in other words, yes there is "some kind of STD involved here". Oral transmission is a good bet; some cases of NGU in men are believed to be acquired that way, perhaps caused by bacteria that are entirely normal in the mouth and throat but cause problems in the urethra. There is no reason to suppose you are at risk for any infection from a partner's oral bacteria, but there hasn't been any research in this area so I can't say for sure there is no risk.
From those response, I hope you no longer "find it hard to believe" that your partner's NGU occurred in the absence of recognized STDs like gonorrhea and chlamydia. This is anything but "rare". It happens all the time -- as I said, it is the single most common disorder seen in STD clinics.
Bottom line: Don't worry too much about this; it is unlikely either you or your partner will have any serious health problems from it. Take the treatment as prescribed, then move on. All will be well.
Best wishes-- HHH, MD