Things like「どういたしまして」, someone used「と申しまして」to their name, and more that I can't recall at the moment, it seemed like it's a te-form of masu-form, but I can't find any explanations of it. If it exists, how does one conjugate masu into its te-form and negative te-form? What and how are they used?
The short answer is yes. It is rarely used, but 「と申しまして」 is certainly an instance where it could be.
From Martin's Essential Japanese (outdated but often extremely useful), a section entitled "SPECIAL INFLECTIONS OF -MASU" (accent marks removed, -bracketed expressions are mine):
The polite ending -masu is really itself a verb which is used only when attached to other verb infinitives. In ordinary polite speech it is inflected only for [... -masu, -mashita, -mashou]. But in honorific speech, -masu is inflected for all categories except the infinitive [that is, there is no -mashi]. These polite forms are used at the end of sentence fragments, and also in the middle of sentences instead of the usual plain forms, to make the entire expression a bit more honorific:
- Soko e oide ni narimashite... You go there and... [in normal speech this would be soko e itte]
- Odeki ni narimashitara... If you could do it... [in normal speech this would simply be dekitara]
- Yukkuri hanashite kudasaimase. Please speak more slowly.
In all these examples (and another, longer one I omitted), it's being used only with the very honorific-oriented o[verb -i] ni narimasu and kudasaimasu verbs. This is no accident. Aside from the fact that it comes smack-dab in the middle of a lesson on honorific speech, this is exactly the context in which one might use it (these two verbs aren't the only one it can be applied to: but they would have a similarly high level of formality, as moushimasu does in contrast to iimasu or yonde imasu. This level of honorifics is pretty much reserved for addressing an Edo-era samurai, or perhaps the CEO of a very large and powerful corporation (and even then, probably more likely in an anime than in real life - the "powerful old family" or zaibatsu kind of situation).
However, there are some holdouts where you'd see it (outside of a movie or the like). 「どういたしまして」is certainly an example, but is of course a set phrase. Still, it's used with someone you're either deferring to, or you're only beginning to be acquainted with, as is 「と申します」, and if you were to conjoin that expression to something else that follows it (or perhaps just to give it one of those "softened endings", like ending a sentence with が or けど?), then changing it to 「と申しまして」 is entirely appropriate.
But yeah, there's not really much call for it in "real" situations, and basically no cases where it'd actually be required of you to form such a phrase.