My dog (90 pound German Shepherd) came down in August. I used United PetSafe, which is more expensive than most airlines, but, on the U.S. side, seemed like a better service and I could get her on a non-stop flight, which I wanted.
Make sure you get the international vet certificate, which must include an indication that the dog has been treated for internal and external parasites. They will also need the rabies certificate, with the ID number of the vaccine administered.
If you use PetSafe, they won't take a kennel with plastic fasteners, so I had to replace those with metal screws and wing nuts. Make sure you have the two containers for food and water attached. I used both of mine for ice, which they gave me at the terminal.
As for the temperature restrictions, UA didn't seem to concerned with those, because she was not a snub nosed dog.
While I recommend PetSafe, know that if you go this route, it's more expensive and, depending on when you call with questions, you may get UA Cargo people, who may give you conflicting info as to the travel requirements. In one case, I was told I needed a certificate from the USDA (I called the Dominican consulate, they laughed and said "we stopped requiring that years ago.").
You also need to know what size of kennel you will be taking and specific type of aircraft that will operate on your intended route. There are size limits for specific aircraft. Also, there are travel embargo periods for dogs due to the high demand for checked luggage space on the aircraft.
The actual travel process is pretty straight forward. I checked in at the PetSafe terminal in Chicago early in the morning (5:00am). At the terminal, they will verify all documents, confirm that the kennel meets requirements, and weigh your dog and calculate the final shipping price (her one-way ticket was more than my one-way ticket). The were very friendly, professional, and "dog people."
I ended up sitting directly above the forward cargo door, so I was able to watch them board my dog, who rode directly below me. Just like clockwork, a van pulled up near the end of loading the luggage. As soon as the luggage was on, they got her out. They were petting her through the kennel as they boarded her, which was cool to watch.
Once in the DR (PUJ), the process was a little different. She was brought into the luggage area on a dolly, sideways, which was loaded with the rest of the stuff I brought. It was quite a sight. There is a fee ($10 or $20 US, I don't recall) once the paper work was inspected. She was also very, very thirsty, so have water or have a plan to have someone go to the bathroom to get some as soon as you can.
I was also told that there was another fee ($75), if the shipment has an airbill number associated with it, which United apparently does do (PetSafe is part of UA Cargo, so they treat the dog as a cargo shipment, not luggage). The document guy, who knew we were coming because I had been to the UA office at PUJ they week before to ask about the process, said "here's what we're going to do: I am going to help you out with all of this stuff, we're going to walk with a purpose past customs, if they say anything, just smile and keep walking." So that's what we did. No $75 fee.
Finally, if you do United, do yourself a favor and don't search too hard for anonymous reviews of the service. UA does a lot of international shipments for members of the military (US mainland to Guam, Japan, Hawaii, etc.) I saw their schedule for the day we flew, which I was told was a slow day: 14 dogs flying out of Chicago (a more typical day may see twice that I was told). If your dog is healthy, it's very safe.