In general, we like the concept of arrows. Specifically:
Entry #9 - We feel in this design there's a bit 'too much' going on (so many arrows in so many directions...). The arrows are not indicative of the concept of 'back.' We very much like the 'racing stripe' through the middle of the type. Random idea, is there a way an arrow can be incorporated into the 'racing stripe?' Perhaps to convey the idea of a pet coming back to you through the word? Kind of like this: <--PetBack- ? Maybe with a paw on the right side?
For Entry #10 - We like the hint of a pet collar, but in this design it's tough to tell that it is in fact a collar. What if the collar were clearer and circling the text with an atomic-type swoosh? Also, I think the navy is too dark, but in general do you have any ideas on how to better convey the concept of the collar (by either changing the color or shape)?
For entries #43 and #44, we decided we actually like the text in the middle of the collar. We also like the arrows from entry #10. Could you incorporate the front of the new collar along the arrows from #10 and the text in the middle? Also, we would love to see a rev of entry #9 if you are so inclined.