As kids, we always called them “pop ice” because that’s what it said on the box.
I was about to chime with Sunnyboys. For years and years the only place you could get them where I grew up was in showbags from the show each year. So they were a once a year treat, that the family would have massive fights over if you stole someone else’s 
[I have noticed they are now carried in the supermarkets though]
On the ‘Freezies’ question, I only ever heard them called Funny Faces (from the brand name)
What show? What showbags? I’m afraid I don’t understand.
Annual fairs or carnivals in Australia, like the Royal Easter Show in Sydney. Kids buy bags of cheap novelties and sugary treats. Every Aussie childhood comes complete with memories of bags full of disappointing toys and a subsequent sugar hangover.
Otter Pops exclusively. I can’t remember buying other brands, but they’d still be the same. It’s the same issue as if you ask me for “facial tissue” when everyone else says Kleenex. You’re technically right but that doesn’t mean I won’t roll my eyes at the specific pedantry.
Icee is the brand name for polar bear-themed drinks, usually in blue or red forms. Slurpee is a similar version by 7-11. Squishee is the Kwik-E-Mart version. You drink all these through a straw.
Another Aussie here They are called icypoles around my way
I believe ‘The Show’ in Aussie vernacular is similar to County or State fairs in the States. As to showbags, never thought that they were uniquely Australian. They are predominately aimed at kids and are full of cheap toys, chocolates, and lollies (candy).
Here in Northeast Ohio, we always called them freeze-pops, regardless of brand.