As an Elite Partner, I only ever have to use my card at partner lounges and I doubt that flashing your card physically would make you a EP by NZ's system. Fo NZ lounges I just scan my ticket, so my guess if they are cancelled in the system, their AP number won't be associated with EP and they would be denied access. It should show in their AP account if they have EP status still.

As for specifying someone new, it doesn't matter what tier they are and when you nominate them - their EP status anniversary is your AP anniversary, not their own. Bare in mind it takes 4 weeks for them to send out EP packs, so on the day you cancelled they would have already sent your new packs

Hope that helps (and I'm accurate)!