You should know that the laws that protect service animals don't have much meaning outside the states. Whereas in the US you can essentially get seating with a service animal with no questions and no fees, this is no guarantee overseas.
I would recommend making sure that your flight overseas is direct to Sweden, you basically only need to deal with the pet issue while boarding a plane. I fly direct to ARN on UAL from the states plenty with my cat. When I transfer through another European country I often have to deal with new fees regardless of what I've already paid for my US flight even though it's the same ticket, and the fees in Europe can be astronomical, I've paid more for my cat to fly than myself in some situations. Some airlines charge for pet carriage by the kg, though how they can justify that is a mystery to me.
I have discovered that if your animal is quiet and if you are conspicuous enough, then you can usually get on planes without any questions, especially now that security is run by TSA and not the airlines (which is where they'd usually turn you around to get a receipt for the PETC). My cat has travelled enough that she doesn't make any noise without any medication, to the point that I've gotten off of international flights and the people seated next to me were shocked to see that I had a cat the whole time. Though be prepared to move if anyone has allergy issues, of course, I found myself in the back row of a plane once (no recline, ouch!). You don't have much recourse if you don't have a PETC reservation.
If your animal hasn't travelled much, then I highly recommend talking to your vet about Acepromazine. It's not a tranquilizer (which is dangerous for animals on planes) but does a great job of removing their anxiety - I eventually weaned my cat off of it over her first flights with much success and a cat who was once freaked out to go in the car now has more miles than most people (if only the airline honored them
.
Last bit of advice - if you're going to/from Europe with your animal, then get an EU pet passport the first time you're in Europe. Big help. Otherwise you need to get a round of shots and the like every time. More info here:
http://Daveola.com/Zoo/Travel/USA-to-Sweden.html