1) Definitely spot at Kingsford Smith when you're in Australia; check out this
website for spotting locations, etc. As far as New Zealand goes, I've never been to the North Island, so I don't really have the authority to suggest Auckland over Wellington or vice versa (those are the obvious contenders in terms of spotting). Auckland would probably make the most sense, as it's the larger of the two airports and you'd most likely be flying in there yourself.
2) Fly Qantas. Period.
3) It all depends how much you want to get around with the time you have. For Australia, I'd definitely say see Sydney (the Opera House, the Harbour Bridge,Taronga Zoo, ferry to Manly, excellent restaurants, etc.), even if it's for one or two days only, the Great Barrier Reef, the Red Centre if you can get over there (Uluru/Ayers Rock, Kata Tjuta/The Olgas, King's Canyon, etc.), the Kimberleys and the Cairns area (again, if you have the time or want to fly more), Melbourne, and maybe Canberra. As I said, it all really depends on how much you plan to get around and fly.
As for New Zealand, I can only really talk about the South Island (which is definitely worth a visit, especially if you want to get away from big cities). My absolute favourite place down there is Wanaka - not just for the skiing, which is just about over now - and you might want check it out. It's a small town on the banks of Lake Wanaka which I just find very calming and beautiful. There are some excellent restaurants down there as well, so you may want to have a look. As far as South Island attractions go, I'd suggest: Milford Sound (either by car, which can be a bit trecherous at times, or by sightseeing plane - very
spectacular!), the Glaciers (Fox - the slightly less touristy and nicer one - and Franz Josef, both of which you can take helicopters over/onto), Punakaiki ("Pancake Rocks") and the blowholes, Christchurch, Nelson, the "TranzCostal" train from Christchurch to Picton, etc...
Any other questions, feel free to ask, and have a wonderful trip!