1. I do not know every student.
[I know some students, and I don't know some students.]
2. I do not know all persons in the room.
[I know some persons in this room, but I don't know the other persons.]
3. I do not eat every fast food.
[ I eat some fast food, but I don't eat other fast food.]
4. I do not to go the gym every day.
[I go to the gym sometimes, but I do not go to the gym sometimes. What about my explanations, the pattern [not - every] shows partial negation, dosen't it?]
Yes, you’re right.
I’d use the ...some ... other ... phrasing forall these sentences, just as you did in sentences 2and 3.