1. Anybody does not sleep well during the night.
2. Not anybody sleeps well during the night.
3, Nobody sleeps well during the night.
(Which ones are correct? Are they the same?)
4. Not all sleep well during the night.
5, All do not sleep well during the night.
6. None sleep well during the night.
(Are both the same in meaning? Do #4 and #5 have the same meaning?)
The first one is incorrect. The second one is quite awkward. The third is grammatically correct but seems incomplete. (Surely you don't mean "Nobody" in the whole world!)
The fourth and fifth have about the same meaning and are correct. The sixth is grammatically correct, but also seems in complete.
For three and six, you'd need to add some kind of limitation for Nobody or None. Examples:
Nobody in this dorm room sleeps well ...
None on our sleeper car slept well last night.