1) Yes. Need to show onward ticket when checking in for your UA flight. As thadocta mentions, if the Qantas flight is a domestic one you will need to claim bags after immigration and take them through customs. So in that case little to be gained by through checking.

2) With United's track record on Oz flights (check out the thread in UA forum), you will want several hours. If their on-time performance improves to that of other airlines then 90 minutes for international to international (QF flight numbered 1-399, and yes there are some SYD-CNS) or 150 minutes for international to domestic are comfortable.

3) Qantas are not obliged to unless you are on a fully flexible fare. In practice they probably do whatever they can to get you to CNS in the event of a misconnect. But if the flights are full you might be waiting a while (and/or routing through BNE).

4) As with most other airlines not based in USA, carry on limits are stricter. They may be enforced at check in, at the gate or by staff operating security checkpoints. In practice as long as your carry ons do not look excessive (by Australian standards not US ones) they will not measure or weigh them.