Yes you could take on a flight companion animal just part of the way (by short-checking the crate). Just plan enough time around connecting at the layover airport because you'd need to retrieve the animal from baggage claim, pass the French customs/agricultural inspection with the animal and exit into the arrivals hall where'll you be greeted by the anxiously-waiting new owners. Plan at least a two-hour layover, more is better.
These days, people use social networks- mainly FB groups - to search/find flight companions, There's hundred of groups out there, and things are pretty fragmented, but the people usually are networked pretty well.
Search FB groups for flight companions, flight buddies, flight volunteers... example: https://www.facebook.com/groups/504130289622324/

It is rare, but there are offers to pay for the ticket, so yes, it's possible to do a mileage run while doing something good at the same time .
For example, here someone is urgently looking for someone like you who'd fly a dog from Thailand to NY and offered to pay part of the ticket: https://tinyurl.com/yc8zwbr5