I think rules have changed lately. Because American Airlines website now says pets can only travel up to 12 hours.
A friend of mine did it, you can do it by yourself. I think these agencies do a little bit of overkill to charge extra. I have another friend getting pet passports and all that crazy jazz.
You should book a non stop flight and call the airline to arrange the pet fee. Get the approved crate. Get a rabies certificate, microchip the dog and usually you need some health certificate from the city/county/state that you live in (they need to be recently, there are some rules on the dates). I don't think here's that much more to it. My friend said on arrival to Hk you have to pick your pet up in cargo and no one knows exactly where it is so you get the horrid I don't know where little thompper is.
Some airlines only charge a couple hundred US so look around. Or you can hire an agent but I don't see how it helps. You still have to run around here and there.