
All that is required to get your dog into the USA is up to date Rabies and most airlines require a fit to fly certification (can be noted in the pet passport).
If you go with American Airlines you can do the multi city option which works out cheaper than two singles if departure and return flights to different places (you can only fly BACK to UK as Cargo, so much better to return to the continent and drive/arrange a lift back).
Regards temperatures, you can't fly if any of the ground temperatures on stop offs is above 85'F or below I think 45'F (7.7'C). You can fly down to 20'F (-7ish) if you get an 'Acclimation Certificate' from your vet that has to have specific data on it.
American airlines have a good page giving details.
I would whenever possible opt for a direct flight.