Welcome back to the forum.
No STDs are transmitted by kissing, regardless of either partner's recent exposures to other persons. Neither your kissing partner's methotrexate treatment nor her underlying health disorder would increase the chance she would have an oral STD infection or have any effect on transmission by kissing. As for your symptoms, the cause is obvious: you caught a cold. Kissing doesn't transmit STDs, but it is the most certain way to transmit a cold or any other respiratory virus. Or you could have been infected from anyone else in your environment.
As for your specific STD question: Chlamydia rarely infects the throat and is rarely transmitted by oral sex, and never known to be transmitted by kissing. Gonorrhea can be in the throat, but not known to be transmitted by kissing; and usually doesn't cause sore throat. (The vast majority of oral gonorrhea infections are entirely asymptomatic.) And no, gonorrhea does not travel through the body to show up at additional anatomic sites.
I recommend against STD testing, assuming this is your only potential STD exposure. You can just forget this incident and move on.
However, before we end this thread, I am intersted to hear the outcome your questions on atypically prolonged molluscum contagiosum, which you asked about on this and the dermatology forum 5 years ago. I assume it cleared up and you stopped having new lesions. How long did it take?
Regards-- HHH, MD
Kissing always carries potential risk for HSV-1, but oral herpes (transmitted orally) is not considered an STD and the risk is low if she did't have an overt cold sore. Syphilis could theoretically be a risk as well, but the chance is sufficiently low to be ignored.
Thanks for the follwow-up on your molluscum.
Thank you for your advice - I probably let her choice of kissing partners (gay men from musical theatre) and the persistence of this sore throat worry me more than I would have otherwise. Plus I've been allergic to the last 3 antibiotics I've taken over the years.
I assume you didn't mention herpes (or syphilis) as it did not apply either. . . .BTW - the molluscum took somewhere between a year and a year and a half to resolve completely.
Thank you ...