I have been debating traveling to St. Thomas Virgin Islands and spending any where from one to two years there. My wife was raised on the island and has family there. But my major problem is my birds. I have done quite a bit of research on this for that particular area though.
Alaska doesn't need a health certificate since it's part of the US. BUT, you will have to check with the Alaskan agriculture Department to see if they require something. Your avian vet should do a health check within 10 days of your flight. This you can verify with your airline. Some airlines allow birds to travel in the cabin and some do not, like American Airlines, some have a limit of pets that can travel in the cabin under the passenger's seat, Delta I think is like 4 if I remember correctly (this is for the whole cabin) so you MUST contact them before hand and get to the airport at least 4-5 hours before your flight.
Yes, you will have to pay extra and it averages about $125.00-$150.00 The carrier must be airline approved and it has to be small enough for it to fit under the seat, but large enough for your bird to be able to stand up in. There is also weather restrictions or time of year. Most airlines will not allow birds to travel as cargo in the summer months.
Now at the airport, you will need to be there 4-5 hours ahead of your flight for security check points and to make sure and check in early in case that there is an issue. The number of animals allowed per flight is a first come first serve basis. So the earlier you go the better off you are. Liquids are not allowed. So you will want to freeze grapes or fruits that the bird can get liquid from. This is a biggie here: MAKE SURE YOUR BIRD IS USED TO THE CONTAINER, this is very important to keep your bird from stressing out and becoming noisy and also because you will more than likely be required to remove your bird from the container. They can remove you from line when they see that you are carrying a pet container and take you to a nearby room where they will ask you to remove your bird so that they can inspect the container. So you must be very comfortable with handling your bird. I read one article where this happened to a traveler and he had a heck of a time getting his bird back in the container after the inspection because the bird was soooo excited he kept wanting to bite him.
You also might be required to show ownership of the bird once you get ready to return from Alaska to ensure that it was bred in the US and not imported. So that is a major issue to find out about like how would you go about doing this unless you have the birth records, sales receipt or so forth.
I hope this helps you out, if I left out anything, I'll add to the post. Good luck with your trip.