Correct way to omit words?

I think there's a lot of different stuff going on here: some of your examples, for instance 'Come what may' and 'if need be' are simply idiomatic (check in a dictionary); others I find very dubious, for example 'Be that true' doesn't sound right to me at all.

I think we often use a kind of ellipsis with 'why', as in 'Why the long face?', although, again, I'm not very sure of the naturalness of your examples.

The last two concern 'have' as a main stative verb (e.g. for possession). Here we have a choice (at least in BrE) between the 'Do(n't) you have ...?' and 'Have(n't) you ...?' constructions, a choice we don't have, however, when 'have' is a main action verb, as in 'Did you have breakfast this morning?'

My information (Swan - Practical English Usage) is that the 'Have you ...?' construction is used more in BrE, although it is often seen as more formal.So even in the UK we're probably more likely to say: 'Do(n't) you have' or 'Have(n't) you got ...'. Having said that, your two 'have' examples sound just fine to me. Which gives us three possibilities.

Without getting into a debate as to whether or not "why" is a complete sentence, I would say that "Why the question?", etc., works as a shortened form, specifically because just plain "Why?" stands on its own.

As for "Have you any idea...", this isn't a shortened form at all. 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.

Moreover, if ‘Haven't you anything better to do?’ when meaning ‘Have you not anything better 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’?

porche, referring to your last paragraph, does that mean ‘Had he breakfast this morning?’ is correct when meaning ‘Did he have breakfast this morning?’ I just want to be sure.

Correction: ‘porsche’ with ‘s’.

Warsaw Will: while I agree that "Be that true" does sound quite incorrect, what about the phrase "Be that as it may..." which I hear often?

If you are criticised for leaving words out in what is a well-known and commonly used expression, owing to its being ungrammatical (eg, lacking a subject or a verb), just say it's ellipsis. "Okay?" (= "Is that okay?"). That should put their gas on a peep for a while.
That is why commands or orders don't have subjects: "Stand up!" - who is standing up? It is understood, so left out.

In the examples of phrases using "be" mentioned above, surely we have instances of ellipsis and the employment of the subjunctive because we are talking of things which may or may not be. As in "Be that true", which suggests maybe it is, maybe it isn't. It is short (ellipsis) for "If that is true (and it may not be)".

Makes Japanese seem easy by comparison, doesn't it? [ 'It' is missing from the last sentence, so no subject because I assumed it: I used ellipsis. We all do. Often. Especially in conversational English. Look! There I go again.]

you is all rong