1. SYD airport has international transit facilities, so you can stay airside if transiting the same day. You cannot stay airside overnight.
2. ETA is required if you go landside (clearing immigration and customs).
3. You can transit with 2 different tickets. But Air NZ has strict rules about when it interlines on separate tickets -- to encourage you to buy NZ ticket for the whole trip; so not sure if it will interline with UA on an award (you can check with NZ about this).
4. SYD has a transit desk that can issue BP without going landside, but if your bags are not interlined, you will have to clear immigration/customs and checkin with UA at SYD.
5. SYD airport is closed for a few hours each night, and security does a sweep, so you must vacate the airport building for at least a few hours. Rather cold in SYD at night this time of the year (not freezing but close enough). Doubt you can stay in the train station, and going into town is about $30 round-trip on the train or shuttle; can do it by bus for less, if no bags, but overnight bag storage at SYD is expensive (lockers have been closed for security reasons, so there is a rather overpriced baggage storage shop there instead, cost is based on size but will typically cost $40 for a day).

Cheapest would be to find NZ award space, it shouldn't be too hard if you have route flexibility!