Yes you can enter the FF number on website booking. You set up a profile and can enter multiple FF # if you want and a default. At time of booking either accept default or enter new.
The NZ services are on the 2 class 777 (or 3 class 747).
Snacks are of chippie and choco bar variety. Can also get fresh fruit and drinks.
If for any reason you dont end up checking in online, you can check in for both sectors at LHR - no need to visit transit desk in SIN for BP.