Is pelvic inflammatory disease something that could flare up and clear up without antibiotics and then suddenly recur again a month later?

Probably yes: In the pre-antibiotic era, most infections cleared up without treatment, but many would flare up again later. I am unaware of reports of this with PID, but probably it can happen. If so, it may be especially dangerous in regard to risk of infertility and tubal pregnancy. Repeat PID causes more tubal scarring and its consequences than single episodes. If you suspect this, get care ASAP. Good luck!