One of my pregnant Puma's just gave birth to a girl with deviant coloring. Does this do anything? What does that really mean?

A deviant coloring is just a skin pattern as far as I know. It doesn't really do much else.

Deviant color is a rare genetic mutation that rarely happens. Go see the list of skins in 'help' to give yourself a sample of possible deviant colors. A deviant color only means that pet has been born with a highly rare mutated color and/or pattern. Many players consider deviants very valuable, and they are really hard to breed. Consider yourself lucky to have succeeded!

For the dog stiuation, I would grab the bat, throw it far away, and when the kid was occupied with that, grab the dog and run.