Sometimes, the character is a "true were", a creature that can only change from human form to one particular animal form, like Faythe, the main character and werecat in Rachel Vincent's Shifter series. In other instances, the character is a "shifter", a creature that can change from human form into any other animal form of their choosing, like Sam Merlotte in Charlaine Harris's Sookie Stackhouse series.
In all of the were and shifter books that I've read, one theme seems constant. No matter what animal (or animals) the shape shifting character can turn in to, the characteristics generally associated with that animal are true of the character in their human form as well. Sam Merlotte, for example, is a loyal, steadfast man who is devoted to those he cares about. The form he takes most frequently (though he can adopt any form) is a dog. Not just any dog, but a collie. The favored animal form represents the man seamlessly.
Above: Sam the man
Below: Sam the dog. Notice, he's still wearing flannel
I don't have an answer for the first question, though I'd love to hear some theories. As for the second, if I were to suddenly find myself with the ability to shape shift, I would definitely be a cat. Not a big, exotic cat, sadly. No, I'd be a common house cat. I like to be pampered, but only when I want the attention. I'm finicky. I'm not particularly fond of water. I have moments of crazed activity followed by hours of napping in the sunlight. I can amuse myself for hours on end with nonsensical stuff. And, I'm secretly bent on world domination. ;)
What would you be?