Most of Australia is actually kid friendly. Since you are likely to rent a car, you wont be doing much walking unless you want to spend most of your time in the cities. The only area which I think is not suitable for a 5 year old is Gold Coast. While others might not agree, I do not see how a kid at that age can really enjoy/maximise the rides.
Of the multiple trips to Australia, I have only done once with a then 4 year old kid. We went to Western Australia, i.e. Perth, Albany and Margaret River. I found the trip to be a relaxing (perhaps cos I know the area well) and my kid did not give me any problems when she was forced to strapped in for hours in the car. The good thing about Australia is there is always some interesting sights and areas to visit a few hours from any major city that you are visiting so best is for you to find out what are the sights/attractions you want to visit before making a decision.
For cooler weather, the best months are June - Sept with end July/early August being the 'coldest'. Do note that the drawback of visiting during this season is obviously shorter daylight hours.