Does the Australian head-of-state live in Britain 🇬🇧? If true, why is Australia unable to have a resident Aussie as its head-of-states ?

The Queen let's Australia run itself.

It won't be for ever; days are indeed very limited.  It was no so much as people voted in the referendum to remain subjects of the Queen (of Australia), belonging to the Commonwealth, as it was the model presented to us at that time.  We are in no hurry for a republic like what you have seen under Trump's reign.  Case of best of two evils for now.  

What are you going to do with this obsession when Y/A shuts down? 

Australia can't have a resident head of state because Australia has voted to remain part of the British Commonwealth.  And therefore to remain subjects of the British monarch.

By definition, that monarch is the head of state.

Australia is "able" to change that, they've just chosen not to.