Either way would work, with very little difference in meaning. (“Have” is probably an action verb in this context, meaning to consume the meal.) That being said, we need a direct object, in this case, “it”. You probably didn’t hear “it” because the vowel is reduced to a schwa, and in Australia, the T was probably reduced to a glottal stop (pronounced in the throat). Note that the whole sentence requires a subject (like “you”) and an auxiliary verb, though these are often omitted in speech: “Will you have it here or to go?”, “Are you having it here or to go? or “Will you be having it here or to go?”

I had to laugh a little when I read this question because this is one of those things in the English language that differs from place to place. I was a little confused the first time somebody asked me about "takeaway". They say "takeaway" in Australia and New Zealand and probably in the UK. We say "take-out" or "to go" in the USA and probably most places in Canada. The opposite is "eat in", "dine-in", or just "for here". I think that "for here" will work in Australia but you should wait for an answer from somebody who lives there to be sure.

Both are fine. 'Having' is using a continuous tense, so indicates something that takes a longer time. As having a meal takes a little time, then it fits both the longer, continuous tense, and the shorter 'standard' future tense.

You can say “have” if they say “have”. Otherwise, you can say “having” if they say “having”. I live in Sydney, and the standard here is also “having/have here or takeaway?”. Sometimes, they do just say “having/have here?” without the takeaway option, which is actually common too. Has this ever happened to you in Melbourne? In occasional cases they sometimes say “takeaway or having/here?” or just “takeaway” by itself.

Neither of them are grammatically correct - that's what happens when you're in a casual speaking environment. But "having" is better because it sounds like shorthand for "are you having it here?" You can respond with "here, please".