politeness levels of shortened grammar

[Dehitay]

I believe when you shorten 待ってください to 待って, it's still considered 丁寧語 though I could be wrong about that. However, it's still polite enough to be used in conversations where you would usually use 丁寧語.

In specific, I was wondering about 2 grammatic structures for which I'll give examples
shortening 行かなければなりません to 行かなきゃ
shortening 勉強しなくてはいけません to 勉強しなくちゃ
Would these be considered 丁寧語 and if not, would they be appropriate to use if you're speaking to your teacher for example.

[Yudan Taiteki]

I believe when you shorten 待ってください to 待って, it's still considered 丁寧語 though I could be wrong about that. However, it's still polite enough to be used in conversations where you would usually use 丁寧語.

That is incorrect. 待って is significantly more blunt and less polite than 待ってください, and is not appropriate to use in cases where 丁寧語 is required.

In specific, I was wondering about 2 grammatic structures for which I'll give examples
shortening 行かなければなりません to 行かなきゃ
shortening 勉強しなくてはいけません to 勉強しなくちゃ
Would these be considered 丁寧語 and if not, would they be appropriate to use if you're speaking to your teacher for example.

No, they are not 丁寧語 and would not be appropriate in speaking to a teacher unless they are embedded in a larger clause -- in this case it would almost have to be a quotation (i.e. 行かなきゃと思って、走り出したんです).