I've been researching it myself since I'm moving to London (from Israel) next year with my dog. She's rather small, but exceeds 8kilograms, which is the weight allowed (the weight of the dog + the cage he's in) to bring a dog into the cabin. Otherwise the dog flies in cargo, and that freaks me out.
There's the temperature issue in the cargo, and oxygen too. I know some airlines don't fly pets during the heavy summer/winter months cos it gets too warm/cold in there.
I did find a website for an Israeli group that specialize in flying pets. They help you with everything pre-flight then meet you at the airport and when you check in they take your pet, up to the point where they put it in cargo. They also check the temperature and oxygen and call you once the pet is on board the plane. It costs extra, of course, but I think it's worth it and I'm guessing other countries have such a service too (I know British Airways airline has its own service like that).
Other things - Europe do have the Pet Passport program in most countries. Even though Israel isn't a part of it, and even considered a "third world country", all I have to do is a blood test 3 months before the flight, some other tests like tapeworm and rabbies, make sure she has the microchip, and she's good to go.
Certified service/assistance dogs are legally allowed in the cabin on all airlines - or should be on all of them anyway (since, otherwise, it's discrimination against the disabled). I actually tried to check if I can certify my dog as a service dog because of my own psychiatric issues lol but it seems close to impossible (though I heard of people doing that in the US, only to have their pets in cabin with them).