Usually if there's a tera burn as a part of another big event thing like this, it'll be under that event's specific quest list rather than mixed in with the normal event stuff.
Reboot, to my knowledge, does not have access to permanent pets. Someone who actually plays reboot will have to clarify this.
Permanent pets, in general, function exactly the same as your typical 90-day pet, only without needing to be refreshed every three months. Granted they cost a good bit more than these regular pets, but you save money in the long run. I'm not sure what you're talking about with a P on pets in the cash shop, but I don't think that denotes permanence.