Can you remove the ability for someone else to be able to rename your pet with a nametag? Sure you can respawn it but I think i'm going to get annoyed when I turn around and my sheep isn't rainbow anymore.
Also rainbow sheep are named _Jeb instead of {player}'s pet.
Edit:
Im also going to add to this with some new pet ideas:
Slime
Cat (tamed ocelot)
Guardian
Enderman
Baby forms of pets maybe?
Thats all I can think of.
Edit2: Make it so you can ride the undead/skeleton horse c: