Dogs aren't considered people (in the US), so it would be considered stolen property _at best_ if this were an actual thing.

Likely, they'll pass the animal on to a shelter and give you the info as to where to pick them up. It wouldn't be considered "stealing".

To this point, after crossing the state line from Nevada into California, there are highway signs that state the fines for littering and abadoning an animal on the side of the highway. The fine for littering is significantly higher than animal abandonment.

This depends entirely on how quickly they get to the vehicle, but given the money involved, I'm sure they will just lobby to have the death of the dog pinned on the owner.