Are the forms ‘Haven’t you anything better to do?’ and ‘Have you any idea [...]?’ correct/proper/formal English?
Moreover, if ‘Haven't you anything better to do?’, meaning ‘Have you not anything better to do?’, is correct, then, following the same logic, isn’t ‘Hadn’t you been there, they would have fallen through the ice’ also correct when meaning ‘Had you not been there, they would have fallen through the ice’?
As for ‘Have you any idea [...]?’, in another forum, someone answered, ‘It isn't a shortened form of "Do you have any idea..." in English, you may often form a question by adding a preceding "Do", but in many cases, you may simply swap the subject and the verb. "You are..." becomes "Are you...?" "He will..." becomes "Will he...?" "I have..." becomes "Have I...?" This is the normal way we construct sentences, not some informal shortening.’
If so, is ‘Had he breakfast this morning?’, the subject and verb having switched from ‘He had [...]’, correct? If not, why?
Hopefully, all the questions will be answered. The more informative they are, the better.

Thank you in advance.
P.S. How is a user account created here? I clicked ‘Join our community’, filled out all the information under ‘Not a member yet?’, typed in the code and clicked ‘Create Account’, but nothing happened.