If the Hound's Tooth has an injured pilot critical damage on it, but is then destroyed would the injured pilot card follow to the Nashtah pup?

Nope. Someone emailed FFG and their reply was that both damage cards and upgrade cards on the YV-666 are ignored when transferring your skill and ability to the Pup,

which includes if you have Veteran Instincts on Bossk, he loses that when he transfers to the Pup as well. Back to his base PS.