"Don't let" vs "Let sb not"

I can’t understand the difference between “Don’t let” and “Let sb not”. Which of the next sentences are correct? Could you explain what is the difference between using “Don’t let” and “Let sb not”?

1. Don’t let them go there.
2. Let them not go there.
3. Don’t let them watch TV late.
4. Let them not watch TV late.
5. Don’t let’s stay here.
6. Let’s not stay here.
7. Don’t let her forget it.
8. Let her not forget it.
9. Don’t let him be rude.
10. Let him not be rude.
11. Don’t let him read my letter.
12. Let him not read my letter.