Pet Running off/Randomly despawning: This is a bug with the core, and unfixable at this time/at all?
Hit stay then hit passive then hit follow and he will return to you when he runs off (do it soon as you notice it as when it gets out of visual range it despawns). the only random despawns i have had was in booty bay at the very top, pets cant sit ontop of gameobjects, so when at the top the pet is quite far away 9out of visual range) and will despawn.

No Pet Skills at trainer: Used to exist, something broke? On a list of things to fix?
correct

Pet Skills can't be set for auto/constant use? Is this just a visual bug or something not implemented?
visual for claw, rake and cower (try manually clicking itl always say its on cooldown). but growl you see it being used constantly no visual defect on it (make sure you rightclick and can see the frame spinning for auto cast)