Welcome to the STD forum. Unfortuately, this really isn't a good resource for your questions. It is clear you have no STD, which really is the only expertise we offer on this forum. When we have patients like you in our STD clinic, we refer them to gynecologists who specialize in infectious diseases. But I'll do my best.
Itching and burning are common yeast infection symptoms, as you might know, and a positive culture for yeast has to be taken seriously in this situation. What kind was it? About 90% of vaginal yeast isolates are Candida albicans. However, 10% are Candida glabrata, which is resistant to the standard treatments like fluconazole (and all other "azole" drugs). You should check this out with your doctor; maybe appropriate treatment (e.g., micafungin) would help.
Azithromycin and doxycycline are antibiotics, active only against bacteria -- and I am skeptical that a bacterial infection of any kind is the cause of this problem. So I am not at all surprised that azithromycin didn't help and doubt doxycycline will be any better. And you give no reason to suppose your partner has anything wrong or that treating him will help your problem. And repeated potent antibiotics further predispose to yeast and other problems that might be the actual cause.
To the specific questions:
1) I doubt this has anything to do with your hot tub folliculitis.
2) Dermatographia and urticaria are symptoms, not diseases in themselves. But if the same sort of immunologic process involves your genital area, it seems possible this might be the cause of burning, irritation, etc.
3) Your positive culture for Ureaplasma urealyticum probably is meaningless. Although there is a certain amount of controversy around UU and whether it causes any diseases at all, it has never been implicated as a cause of vaginal discharge. The weight of evidence indicates it is just a normal bacterial inhabitant of the genital tract. At any point it time, it is present in the vaginas of 50% of all women.
4) Three possiblities for additional medical evaluation come to mind. First, as suggested above, discuss Candida glabrata with your doctor(s). Second, try to find a gynecologist who specializes in infectious diseases. Such persons often can be found at major medical centers or medical schools. Finally, if those approaches dobn't pan out, in view of the possible association with urticaria and dermographia, consider seeing a dermatologist, allergist, or both.
In any case, there is no reason to suppose any STD here.
I hope this helps. Good luck-- HHH, MD