As a NZ citizen, I'm not allowed to vote in Australian Federal elections, and I gather, state ones too.
In the UK we were able to vote in local elections, and London mayoral ones. Are there any elections in Australia that non-citizens can vote in?
No. Rules regarding eligibility to enrol to vote are uniform across Australia. Only Australian citizens can enrol and vote federal elections and referenda, and state and local government elections.
The only exception is certain "eligible British subjects" who were already on the electoral roll on 26 January 1984.
Yes, you can apply in (most?) areas to enrol in local goverment elections if you are a ratepayer. Often you have to live in Australia as well.
You may have to re-enrol annually, and you will get fined if you enrol but fail to vote.