I stayed with my 3-year-old boy and my parents. We enjoyed going for a walk to beautiful Carlton garden and Lygon street like a resident of heart of Melbourne city! For family with small children, I recommend to visit Collingwood children's farm, which Celia recommend us. My son loved it and visited there 3 times!
Celia and her husband are very helpful and reply quickly, so you don't need to worry about anything.
But particularly for Japanese family, you have to be careful not to run a bath, because tank of hot water isn't very large.
Anyway, we really enjoyed Melbourne life without any anxiety. Celia, thank you very much for your support. We love Melbourne and your cozy apartment!