I'm in Sydney at the moment, two thirds of the way through our Australia trip. You will not need boots for either Ayres Rock or Cairns - trainers will be fine. You are in for a real treat - this has been an amazing holiday so far, with each day bringing something special. Don't worry about bringing enough clothes for the whole trip. Each hotel has a very cheap and efficient guest laundry and you have sufficient time in most places to get it washed, dried and ironed.
Sally