First, your question would be easier to read if you went a little easier on the italics. Also, it's good to get in the habit of using a neutral tone, even when you're feeling confused or frustrated.
It's fine to send some queries by email, but try to select the most likely people to send your queries to. You could start with one per program that you're interested in, but be ready with a second name in each department. Wait at least three business days for a reply before trying name #2. Keep your email fairly succinct and neutral in tone. Skip the part about the confusion. (It's fine to be confused, and they're used to that.) Make a spreadsheet so you can be organized about this.
I think you should also ask them what to do about the GRE. My gut feeling is that regardless of which program you apply to, you should take the Physics GRE, but this is something you could ask at Physics SE if you don't get a response from a mathematical physicist here about that particular issue (which you should add to your question with an edit, by the way, since it's an important point). For example, in an email to someone in a physics department, you could say, "I'm also applying to some math programs, which require the math GRE, but the subject test GREs are offered only once per (year? two years? I didn't understand this point) in my country; so I would like to ask whether your department would accept the math GRE in lieu of the physics GRE." And vice versa.
A tricky part of this is that you may get some answers back that go against what your physics professors have advised you. If none of them are very close to you, and won't be bothered by that, fine. But if you think someone might be bothered, then I would suggest you share a sample draft email with this person, and most importantly, confer with him or her about the choice of specific departments and individuals you'll be emailing. That way, once the answer comes back, you'll be able to share it with this person comfortably.
A word about the choice of individuals to send your queries to. The ideal person to write to would be someone who is strong in your area of interest, but who seems to have some interest in student advising as well, for example, with a current or former position in department administration, such as "dean of graduate studies," or "graduate program director," or someone involved with department admissions.
If you are talking about applying next winter for a fall, 2018, start, mention that in your email, so the reader doesn't feel alarmed that you might be talking about applying now for fall, 2017.
(If you are talking about applying now for a fall, 2017, start, then just throw some darts at the dart board as quickly as you can, and don't waste any time thinking, since this is March already.)
And by the way, Johanna is absolutely right ("You should apply to the department the professor is associated with"). In trying to figure this out, you'll have to look at the faculty bios rather carefully.
Note that sometimes you can have, for example, a physics department as your home department, but work with an advisor whose primary appointment is in, for example, a math department. This is easiest to swing when the professor has some affiliation with your department. If s/he doesn't, then you can still check the program's policies, and also it might be possible to get special permission.
You may want to consider:
- What coursework is required in each department? Which program of studies fits better with your interests and needs?
- What about the basic exams? You wouldn't want to get stuck in a math department that requires you to show mastery of number theory, for example. (That's not a realistic example -- hopefully you get my idea.)
Some departments will provide specific advice on their websites for people like you, whose area straddles two departments to some extent.
It could be overwhelming to delve into a lot of departments' websites in depth. If I were you I would start with one that has very clear materials, and in which you are definitely interested. Understand that one as well as you can, and then prepare your first email. If you are able to cite a specific webpage and a quote or two from their site, as the beginning of your question, so much the better. (I.e. show that you've done your homework.)
Last comment. Just as people like me enjoy understanding a question and attempting to say something useful -- there are people in every department who do too. If the first person you try writing turns out not to be responsive or helpful, that's okay, just try again with a different member of the department. You can add at the end of your email something like "If there is someone else in your department I should address these questions to, please let me know."
