do QF crew overnight in JFK?
 

or is it a turn around like many of the SYD/MEL-PER flights are?

When it's operated by an A332 yes. 744 no.

Which base on the A332?
 

Lucky crews 1-2 nights in NYC! Who will be operating the LAX-SYD sector? AKL, SYD, MEL or BNE?

Answer : Yes, AKL SYD MEL or BNE any and all.

Do QF really take their A332 all the way to JFK? That's one long trip in one of those machines - where does it stop? Is there crew rest provided on board?

A332 ops to North America:
SYD (or MEL) -> AKL -> LAX -> JFK

Yes crew rest fitted for both parties.

Isn't it only 4 crew rest seats plus the R1p seat? There's no horizontal crew rest facility available unlike the A380 and 747-400.

Correct, y/c seats only, plus one "high comfort" ( as if ) seat for crew rest. Not really an issue on lax-jfk-lax but certainly the lack of horizontal crew rest for the akl-lax-akl, is.