Is it grammatically correct to say
Don't your father have a mother?
I would prefer
Doesn't your father not have a mother?
Is it grammatically correct to say
Don't your father have a mother?
I would prefer
Doesn't your father not have a mother?
Neither of these are good options, though your version is at least grammatically correct.
Doesn't your father not have a mother?
This is a phrasing that implies you know the answer already and you're merely asking for confirmation. Another way of phrasing this is:
Your father doesn't have a mother, right?
If you're using have to imply "is alive", you're telling someone rather forcibly: "your father's mother is dead". This is sort of rude, at the very least.
The original version, with don't, is incorrect grammatically. But it is better to some degree because it doesn't have the second negation.
Doesn't your father have a mother?
This is a slightly better option - it means:
Your father has a mother, right?
Or, again, if you're intending for have to mean "is alive", it would mean
Your father's mother is alive, right?
Alternately, the version recommended by Versatile and Affordable is less rude because you're simply asking for information, not assuming you already know the answer:
Does your father not have a mother?
In this case, depending on how it's spoken, it can actually show concern.
All that said, all of these are non-standard. When trying to ascertain if someone's still alive or not, we don't say "have". Everyone has a mother and a father (genetically and literally speaking – I'll not go beyond that). If you want to be reminded if someone's grandparent is still living, please be more exact:
Is your father's mother still alive?
Is your paternal grandmother still living?
Or, if you're reasonably certain she's alive,
Your paternal grandmother is still alive, right?
You are correct that
Doesn't your father not have a mother?
is correct since "father" is third person singular.
No, it is not correct to say
Don't your father have a mother?
Because the father is 3rd person singular (=he, she, it) which gets only "does", saying that we can not refer to him in auxiliary verb of 1st and 2nd persons ("do").
The correct sentence is:
Does your father not have a mother?
It is because the structure of the negative interrogative sentence is as follow:
Auxiliary verb > Subject > not > Verb > object
In your example:
Auxiliary verb (=Does) > Subject (=your father) > not > Verb (=have) > object (=a mother)
Take a small tip for such questions: Compare your sentence with: "Does he not have mother" and replace the pronoun (=subject) "he" with "your father" (=subject) and you'll find the answer quickly.
Regarding to the second sentence that you wrote:
Doesn't your father not have a mother?
It does not match the grammar rules of formal English which I studied.