"got" is totally fine. but If you were to use "get" you would have to add the "s" so "gets"
However "gets" sounds a little bit more harsh, kind of like "he forced him mom to clean the room" and "got" although still a bit harsh(depending on context, but usage is fine here) is kind of
softer.
"had" is a little bit different actually. Depending on how you tone your voice it has the potential to be rude, 3a is perfectly fine if said in a normal voice. The funny thing is when you say "get" it makes it seem kind of like, you rule over the teacher, or they are your slave, however it does depend on context.
It comes down all to how you tone your voice, but "I had my teacher correct my paper." would definitely be better in my opinion. I would definitely prefer 1a,2a,3a,4a rather than the respective b's.