OK, one at a time:
I plan to be in Sydney for an additional one or two days (do you think that is enough? And one or two days?)
I suggest you fly in a day earlier. Most people fade quickly after staying up all day on arrival - the flight from the West Coast departure hubs is anywhere from 14:35 hours (from SFO) to 15 hours (from LAX) to 15:30 hours (from Vancouver), and you'll have been up all day already (maybe flying in from elsewhere) since the flights leave the US West Coast around midnight. You'll fall asleep during the second inning... If you go to sleep during the day of arrival, other than a brief nap, you're likely to be turned around, wanting to sleep during the following days and wanting to be active at night when you should sleep...
Lots of people spend a week in and around Sydney and don't think it's too much, so whether you have one or two days makes little difference - either way it's not enough.
Fly to Cairns from Sydney - Visit Cairns and the Great Barrier Reef, along with the rainforest (2-3 days?)
If you take the 6AM flight you get to Cairns just a bit after 9AM and you stand a chance at doing something that day, like taking a boat out or going up to Kuranda, so maybe that day and one more would let you see a minimum, but three days would be better.
Probably drive from Cairns back to Sydney along the coast, the trip could also be done in reverse driving from Sydney to Cairns and then flying back to Sydney.
Not really an option with your limited time. If you never stopped and drove like crazy you'd be on the road for about 30 hours straight, and that would be at rally-speed with on-board toilets and a support system... not realistic. Three days would be exhausting. It's hot up that way, and if you take the shortest route, it's deadly monotonous, and you have be alert to give way to the 'roos and the road-trains, they stop for nothing and nobody. The more interesting route that follows the coast is slower. Either way, you can't win. Fly.
These monsters come at you like crazy, and if you don't slow down and get onto the shoulders, it's not their fault...
I have a total of 9 days in country, not including the day I lose flying in or the day I fly out.
Let's see if this works out - modify to suit:
Day 1: Depart for Australia
Day 2: Arr in SYD, find lodgings, leave bags (too early to check in), go walking, end up at Circular Quay and hop on a harbour ferry (any ferry - the simple fact that you're on the harbour will excite you and keep you awake, but the Manly Ferry lets you walk from the harbour-side terminal to the beach on the open ocean side, highly recommended)
Day 4: See Kangaroos, Koalas (note spelling, and they are neither beers nor bears, they are just Koalas), and other exotic creatures at one of these: Featherdale or at the fabulously located Sydney Zoo where you find many other species also and a great view of Sydney, easily reached by Harbour ferry, or at the walkaboutpark
Day 5: Go up on the Harbour Bridge Pylon Lookout where you have a 360 view. It is easily reached and not expensive. This is not the same as the expensive and elaborate bridge climb - which is an exciting option also of course.
On the way up to the bridge, check out The Rocks, and across the bay is the stunning Opera House - they do tours of the inside during the day, great photo op with fabulous looks out onto the harbour.
There is much more in Sydney, not to mention the environs, but...
Day 6: Fly to Cairns.
Day 7: In and around Cairns.
Day 8: Fly to Sydney
Day 9: Depart from SYD
What other sites along that route would you recommend? Or, am I totally off base and should consider another option? I don't want to spend too much time flying which is why we are looking at driving. Last year, I drove 2 weeks in Ireland so I'm thinking I might be able to pull off a coastal drive here.
Our interestes including wildlife (kangaroos, kuala beers, other exotic creatures land or sea); culture, rainforests, magistic mountains, etc.
I know its not alot of time but that is unfortunately the time frame we have to work with.
Hope this helps.