Questions – Is a Question Beginning with ‘How to’ Grammatically Correct? Analyzing Proper Question Forms

They are perfectly grammatical. That is, they are too grammatical, using an inverted word order to form a question rather than a "do" form. They have become set phrases.

There is nothing wrong with the set phrases "Have you no shame?" and "Have you no money?" but "Has he a friend?" certainly sounds odd. Normally this set form is used with "no", and "Has he no friends?" would be fine.

In normal speech, "Don't you have any money?" and "Does he have any friends?" would be used.

How do you ..? is in fact pretty generic. Of course 'you' could be specific = the addressee, but often it is generic in the manner of French 'on' or German 'man'

Then there is How do we..? which as a generic asks how something is done within a group of people that both you and I belong to, eg. how something is done within the company we both work for.

[Edited section on How do we]