I don't know how it works in Australia, but what google image shows me is not "cukierek" but "lizak" ( a kind of sweets that is on the stick)

Yes. That is what they say in Australia as can be heard in their national anthem "Do You ❤❤❤❤ on First Dates?"

I don't really like it, because it seems too wide, but on the other hand, 'sweets' was accepted already, so okay.

Sweeties is what a child would say or maybe an adult addressing a child.

Yes, Genitive or Accusative singular.