Grammar – ‘Why Do They Don’t Have’ or ‘Why Do They Not Have’?

Your first sentences is correct

They seem not to have learned grammar.

Your second sentence needs to be restated from

They seem haven't learned grammar

to

It seems they haven't learned grammar.

Your third sentence is almost correct, the contraction needs to be expanded because of the "to"

They seem to have not learned grammar.

We don't use to have in a progressive sense when it means to possess: we say I have your phone number, not I am having your phone number. So any version with the progressive is wrong:

• Why I don't have your phone number?
• Why I am not having your phone number?
• Why am I not having your phone number?
• Why don't I have your phone number?

Also, when we make wh- questions in English, we change the word order: the wh- word, like Why, is immediately followed by the auxiliary verb do, then the subject, then the main verb. So the proper order is Why don't I have... and that leaves us with only one choice:

• Why I don't have your phone number?
• Why don't I have your phone number?

That's the only idiomatic, grammatically correct way to say the question.