Would a Gryphon with a lion's body and only wings of a bird be accepted?
No, it should have at least bird head too and preferably bird front paws.

In other words, does a Custom-Skinned Cockatrice have to look specifically like the Cockatrice on-site or be our own interpretation of the mythological creature.
It doesn't have to be a copy of site version of Cockatrice. It can be any creature interpretation which could be identified as cockatrice. The rule is the same: leave the essence of the creature (bipedal bird/reptile hybrid with the bird-like head and preferably though not obligatory with some kind of crest), all other changes are acceptable.

But would it be alright to put scales on a unicorn? And/or give them two horns rather then the one?
Scales for unicorn is ok until it stays a unicorn and doesn't become a kirin. Two horns are ok too, I don't see any problems with identifying a horned horse as a unicorn, no matter how many horns does it have and even it actually stops being a "uni".

is there a breed deffinition for a dragon- cause some species of dragon dont have wings like eastern style dragons/or some water dragons. Are wings a requirement?

Yes, the wings are obligatory. The rule for all species is the same: the number and type of main body parts (limbs, heads, tails) should stay the same. The decorations and details can be any. For example, wings can be feathered but not removed at all.

In regards to dragons: the creature skins should have the same amount of limbs, heads and other body parts with the site versions of pets, no matter if you're using the official base or creating a skin from scratch. If we had hydras among pets, it would be accepted as hydra, but right now we don't have them. The site dragon is a reptilian with 1 head, 2 wings, 4 legs and 1 tail. Everything else can be altered.

So your dragon would have to retain all these body parts. I'm sure vestigeal or smaller wings would be OK.

Here is a list on what comprises each species generally:
Dragon. Reptilian with 2 wings, 4 legs, 1 tail and 1 head.
Roc. Raptor avian. 1 head, 2 wings, 2 legs.
Gryphon. Feline / avian hybrid with bird head, wings and cat hind paws and tail. Front paws can be either avian or feline. 1 head, 2 wings, 4 legs, 1 tail.
Unicorn. Equine with horn on the head. 1 head, 4 legs, 1 tail.
Cockatrice. Bipedal reptilian / avian hybrid. 1 head with beak and preferably a crest, 2 wings or intermediate form between reptilian arms and wings, 2 legs and 1 reptilian tail.
Puma. Ordinary feline. 1 head, 4 legs, 1 tail.