The dictionary hint says that “mám” can mean ‘do I have’, but it is not accepted.
Yes. Mám does mean I HAVE. But MÁM + infinitiv = "should I "verb"" or "am I supposed to..."
It works for all persons.
Máš tam jít. = You are supposed to go there.
Máš auto? = Do you have a car?
The problem is that “have” can also be used as a modal with infinitives in English, but if “mám” + infinitive doesn't mean the same, then “have” should be removed from the dictionary hints in these sentences, and should be replaced by “should” and “be supposed to” to avoid confusion.