What are the Queens responsibilities in Australia, NZ and Canada?

[AnneBullen]

And indeed all the countries of which she is Head of State.
Obviously here she opens parliament, invites the PM to form a Government, speaks to the PM weekly etc.

Does she have to do those things in other countries where she is HoS? If not then why? Is she MORE Queen in UK than Australia for instance?

She is the Queen of Australia, New Zealand and Canada. It's up those individual countries to decide on what she does there. What she does here is irrelevant.

Oh Ok! That’s interesting, I hadn’t realised that. I assumed she had to do certain things as HoS

In Canada her role (opening parliament, speaking to the prime minister or premiers, etc) is performed by her proxy, the Governor General. On a provincial level by the Lieutenant Governor. She is on most of our money, some portraits in govt offices, criminal trials are (still, I think?) The Crown vs Joe the Burglar, the Queen’s Printers does maps etc in our province. When I was a kid we sang Oh Canada and God Save the Queen at all school assemblies, but my kids find this hysterical and assure me they never sang God Save the Queen (it’s a wonder she looks so well without little ones singing for her health).

What are her responsibilities here in the UK?

To be the reigning monarch whatever that entails and I suspect what ever the monarch does or doesn't want to do. Been that way as long as there has been a king or queen anywhere.