When checking if all the students are present in a class, may the teacher ask "Is everybody/everyone here today?" or "Is everyone/everybody present today?" And if not so, must the students reply "No, everyone/everybody IS NOT here today." or "No, NOT everybody/everyone IS here/present today."?
"everybody" and "everyone" are equivalent, see these links:
The normal way to answer in the negative is:
No, not everybody/everyone is here/present today.
As you can verify with google, "everybody is not here" is much less common than "not everybody is here".
I can see there might be a few cases where you would say "everybody is not here". It's more emphatic and could be spoken with hostility. For example, if half the class is obviously missing, someone could say:
Could we get started, since everybody is here?
No, everybody is NOT here. Don't start yet!
Everyone and Everybody both mean "all the people", the question is what group contains the people.
Everybody is "all the people" in the world.
Everyone is "all the people" in a smaller group. This is usually, the people in an already mentioned group, or if no group is mentioned, all the people in the group being addressed.
Many people, even native English speakers, accidentally use "everybody" when they mean "everyone", so be prepared to use other clues to figure out if they really meant "everyone".
When responding, try to use the word "no" only once, and place it close to the verb. This will have the least chance of being misunderstood.
Everyone is not here today.
Is the best answer
Everybody is not here today.
Is an acceptable way to answer "Is everybody here?" Even if it not the correct use of "Everybody", since the person asking used the wrong word, there will be no misunderstanding in the answer, as you are matching your mistake to theirs.