The pumas with different body shapes are in fact created due to a mutation, which has a 1 in 10 chance of occurring when a pet is born with two copies of the same color or pattern gene. As Ascot said, they're called deviants, and they are considered much more valuable than non-deviants. There are also deviant rocs, unicorns, and dragons, although it's possible deviant forms for other species may be eventually introduced as well.
Wild lykos come from a single yearly event where you find the eggs in Explore and activate them by crafting certain items. Unfortunately, that event has just ended for the year. So yes, you would either need to buy an egg from someone, or a lykos that came out of an egg. Unless you were lucky and found one in the pound. If you go looking to buy an egg, make sure to find an Activated Egg, not a Hibernating Egg, because there's no way to activate them outside of the event.
Also, I have this GP1 symurgh with nice genes I'd be happy to give you if you'd like, to get a little head start. Been trying to sell him, but no one's interested. In fact, his mother was given to me as a newbie, so it'd be cool to pay that forward.