Well, if you want to live southside you could try Burwood or Strathfield. Otherwise Balmain or Rozelle. All have good state schools. Hunters Hill is very nice and can be pricey but you can also find some cheaper places there too. It's near North Ryde and is north of the harbour but on the west side, so that might still work for you.
There are more expats in the north of Sydney, but yes, they are more expensive.
Not sure if you have used the cityhobo website yet.
We used it before we moved to Sydney and got lots of very helpful information from it on where to live. It looks at more than 50 suburbs and gives you an honest account of what's what.