The first two are understandable but awkward. Both expressions are really meant for the positive action rather than the negative case (prohibition).
The second two don't make sense. So, in more detail:
"Her mother makes her not keep a dog at home." I would say something like
"Her mother forbids her to keep a dog at home"
Your original expression can be used positively:
"Her mother makes her clean the house every day"
The second sentence, with "has" is similar (to my mind, this use of "has" is more American usage).
The third sentence "Her mother lets... ": "lets" signifies permission (allowance) and it is self-contradictory therefore to make the sentence negative.
The last sentence is a somewhat similar case. You "get someone to do something" - not "to not do something".