"안~ = ~지않다" and "안 하다 = 하지 않다". The first ones are the grammar principle used for giving a verb the meaning of "to do not + verb" (like you did with the second ones "안 하다" and "하지 않다", that mean "to do not do").

You should also be aware of making sentences that mean "I can not + verb". For example: 못 가다 = Can not go.

Hmmm I think those have same forms but different rules to attach it... 안, 지않다, 안하다, 하지않다 all those particular used to say we "don't" 안 is used before the verb root, like 먹다 ---> 안먹다 지않다 is used after the verb, 먹다 ---> 먹지않다 안하다 and 하지않다 take those kind of form too, but with the verb root 하다 which means "do" Can you get my point? It's kinda hard to explain :D But I hope this helps :)