I would say "If the student doesn't do their homework they will be punished." This is as you indicated in a general sense, not referring to a specific person. If you want to refer to a specific person you could say "If he/she doesn't do his/her homework he/she will be punished." Hope that helps!

You should use 'their' and ‘they', because you are not talking about a person of a specific gender. If you say 'his', that implies that you're talking about a specific student who is male.

I think "his", "he" is better, you wrote "does not", so I prefer "his" "he" .

I remember my teacher once telling the class that technically, since you're talking about only one person, you shouldn't use "they" you should say "he or she", but nearly everyone (including myself) uses "they/their" for singular or plural unspecified persons/people.