Upvote:3
American Airlines will NOT check through bags on a separate ticket. This means that in Sydney you would need to clear immigration, collect your bags, clear customs, re-check them, and then go back through immigration/security. You would also need an eTA or visa in order to enter Australia.
Officially, Qantas International check-in closes 90 minutes before flight departure. So fairly clearly, no, your first option will not work.
If you can purchase both flights on the same PNR then you will be able to check the bags all the way through to Queenstown. In Sydney you will simply need to re-clear security and find your departure gate, in which case 90 minutes will be more than enough (and if it's not, due to a late inbound flight, then you will be rebooked due to it all being on the one PNR/ticket). Booking as a single ticket will also save for paying some of the Australia customs/immigration fees as you will only be a transit passenger.