I dont think you can come all the way here and not got to Melbourne. I normally find that there is alot more going on in Melbourne for a tourist than any of the other cities. March 2021 should be the Grand Prix, the start of the AFL and some pretty good grade horse racing. You also have quite a large theatre district with well known shows touring.
I would say your time in Cairns is too long. It is the Blackpool of Australia, and only really serves as an entrance to the reef. One good, but quite expensive, alternative way to see the reef is to go to Hamilton Island. Direct flights from all major cities, nice resort island with some good quality accommodation, and some excellent snorkelling round the island (last time I was there there were at least 20 loggerhead turtles in the bay) and diving a bit further out (about a 1 hr sail from the island).
For the Hunter Valley tours, use this guy: https://gourmetgetawaytours.com.au/about/ The tour is excellent and well worth the money. He will pick you up from your Sydney hotel. Blue Mountains are nice, the Australians love them, but for people from Scotland they are a little underwhelming.
For the bridge, walk over it and go up one of the Pylons, dont spend all the money to see what is essentially the same view from the climb. That money could be better spent. There are excellent helicopter flights over the harbour for the same (or less if you are a group) money. Or you could save your money and do a nice meal. Quay and Aria both have amazing views of the Bridge and Opera House for a special meal.
If you were to cut down Cairns and cut out the Mountain day you would be able to do either Melbourne, or you could drive from Sydney to Brisbane. The stops up the cost from Sydney tBrisbane, such as Port Macquarie, Coffs Harbour, Byron etc are all really nice and all in it is only around 10/11 hours of driving time from city to city