Turkish Airlines – International travel with pets: what to say about the first leg (United, THY)

Since United and Turkish are both Star Alliance, I can buy the whole ticket from Turkish and fly from my small town to a big airport (like ORD) with United, then Turkey (or elsewhere) with Turkish.

I have to do this with a medium sized dog. If I make deals with Turkish, does that mean that the first match with United will also be covered? If United does not normally carry dogs in the country under the same rules (or not at all), am I lucky?

How do you usually fly with dogs with two airlines?