OP, i think you can get the visa online instantly. (i'm australian so i've obviously never had to do it but my american friend tod me this was how he did it).
As far as your destinations, what are you looking to see? I'd say that's an ok itinerary, but to be honest i might skip melbourne if i only had 10 days.
Sydney is great, to me it's got everything and I don't think a first time visit to aus is complete wihtout a stop there. Definitely hit up the beach for a day or two (coogee, bondi and manly are the most touristy ones....there are more further north and south of the city but it will be easy to get to any of those three). After that, there's a ton of stuff to do, anything you can find in any other major city you can find there. Of course you've got all the usual tourist traps (sydney opera house, harbour bridge, centrepoint tower, aquarium, zoo) as well.
when you say 'great barrier reef' are you headed to cairns? If so, definitely try to make it up to the daintree rainforest and spend a night there if you can. (cape tribulation is a little village in there, there are plenty of buses going up there and you can do croc watching tours, sea kayaking, etc). defnitely go out for a day scuba diving - you can do this without being certified if you go with a guide, and it's only about 100 bucks for the day.
Other things to try off the top of my head - fish and chips, a night at the pub getting blind, nightclubs, i actually don't get into cricket much so i don't know if there're any on yet but go to a match if you get the chance (you've actually jsut missed the footy season), surfing.
have fun, i've been gone for 3 years and i'd love to go back for a visit