Araps and the Gramps would have to be great destinations for all of those things - depends on how hot you like it. I have been for a few summer trips and only ever spent one day not climbing because of the heat.
The Blue Mountains is a bit cooler and has plenty of sport, single pitch trad and a few easier multipitches. There are also good cooler crags where you can go on a hot day.
For a bit more exciting single pitch sport and trad you cant beat Point Perpendicular. Plenty of shade from midday at the major areas and if you can't find a seabreeze there then you wont find a seabreeze anywhere.
Summer is the time for Tassie but I've never been - I'm sure others can fill you in.