1. Don't think of it as
resummoning your pet. You're not. When you zone, your old pet is killed. He's never coming back (until you get the AA Pet hold ability). You're summoning a brand new pet every time, so yes, as someone said, each pet would require a new kit.
2.
Quote:
Pet level is determined by the buff spell that you get with each spell level
Not really. The pet is the same level until you get your next pet summon spell.
Unlike pets of some other classes, beastlords get "proc" buffs to put on their pets. Magician pets cast spells, beastlord pets proc like weapons (not a cast and no mana used). Sometimes you'll get a new proc for your pet, without getting a new pet. The pet itself is the same level. If you want to know what level your pet is, have players con him until someone cons even. Viola, that's the level of your pet.
From what I'm seeing on the beastlord spell list, in the sub 50 levels, you seem to get a new pet and it's proc spell at the same levels. Post 50 it looks like you will sometimes get new pet procs without getting a new pet.