Entry #5 is my favorite. I'd like the slogan "your paws are in good hands" emphasized more than the company name, if possible. I like the idea of the automobile being incorporated in the logo. Very nice work.
Take out Prestigious Pet Transportation and put the "your paws are in good hands where Prestigious Pet Transportation is. Can you put a cat in the picture? Also, can you create a logo with a human hand with a paw print in it? I like your work.
Entry #9 - Good, would like to see more of a paw print with 4 pads and the hands to look a little more like hands rather than flower petals, though this is very, very good.
Entry #10 overall is what we are looking for with the paw in the hand and the emphasis on "your paws are in good hands." Still, can you come up with various paw print in the hand for other choices? But this overall simplicity is great!
I feel it's more important to put emphasis on transportation than get caught up in the whole hands theme. There is a point where it starts to look odd and what you want for a logo is to be effective, not odd. Anyway, I hope this isn't odd :) Good luck!