And as you can see, the gifts are wrapped, with the exception of the live animals. (Please, no one point out that this appears to be a mail bag and not a toy sack.)
As per this official picture, DO NOT wrap a dog or a penguin, no matter what your family's tradition has been in the past. This scene clearly shows the animals need to be free to romp in the presents, or even hand out gifts to others. Plus, it makes for a cute photo on Christmas morning to have a giant mouse peeking out from behind the pile of loot!
Let me tell you how we do it around here. Santa brings one especially desired present for each child, like a bike, or whatever tops the child's list. (Bryce bought Cam a HOT DOG MAKER this year because he said he really wanted one. Good thing he didn't want a machete, eh?)
Anyway. The stocking gifts are also from Santa, and there is usually some good small stuff in there. Candy, little games, maybe a CD, jewelry if you're a girl, markers, etc.
And I have usually asked Santa to wrap his gifts because I want the moment to last longer, but within reason. For example, a CD: wrapped. A bag of candy: unwrapped. Necklace: wrapped. Stuffed animal: unwrapped. I figure if each person takes a turn unwrapping their stuff I can get better pictures AND it will seem like they got more things if it takes 10 minutes versus 10 seconds to open it all. But perhaps Santa will go green this year and all the pressies will be naked.
So that's the low down on Santa at the Jones house.
I certainly enjoyed reading how other people do it, and I think that THE REAL ANSWER is this: however you do it in your family is the right way, as long as you stay away from boxing up living critters.