International travel with parrots

If you haven't seen this page yet, check this one out. There seem to be few requirements to exporting cockatiels from the US, because as you said, they aren't CITES-listed.

At a minimum, you'll have to get a veterinary health report usually within 30 days of travel and you'll need this report if you want to bring the birds back into the US. There are also restrictions depending on which country or countries the birds have been in. Right now, most of Asia and part of Africa are on further restrictions due to certain diseases.

You might want to put leg bands on the birds, if not already done, and have that information included in the vet's report or elsewhere. Much easier to prove they're the same birds and may help to recover them if they ever escape.

Also confirm with airlines if there is a restriction on the number of animals traveling on a plane. United limits it to 4 pets in economy, 1 in the other classes and no pets to Australia, Hawaii, or Guam.

Beyond that, this feels like a shopping list question. I certainly can't give more information without knowing what the destination country is. Their requirements will drive what's needed before you leave the US.

1. I would ask your vet, or a good vet that specializes in birds. I think that would give you the best options for travelling. For instance, when we travelled with our cats (Denmark to US) our vet recommended the cargo hold, and no drugs. In her option drugs would stress their systems as much as the travel, and the lower temp in the cargo hold would make them drowsy. In addition, this would allow for a larger crate, they would be crammed in one that fit under the seat. I would not have thought of that myself, even though we have had cats for many years.

2. I am not sure I fully understand the issue, but beyond looking at relevant government agencies in the US/destination country, I don't have good advice. If you have found something specific to cats/dogs on a government website, I would contact that agency and ask where to find information on pet birds.

Lastly, I know there are companies that handle international pet transport, and I am sure they also know about how to handle paperwork. You may look into that as a different option.

even if you're allowed to take them out of the US (don't see why not unless they're listed as endangered in which case even having them as pets might be illegal) you may want to check whether you're allowed to import them to Sweden... Some countries have severe restrictions on importing life animals.

If that's all settled (and all the paperwork properly arranged, better safe than sorry) your best bet is likely a specialised transport company that deals in the international shipment of exotic animals (and they'll likely be able to handle the paperwork as well, for a fee).
You do NOT want to end up at customs with 2 dead parrots in your bag. Not only would your precious pets are dead but you're liable to be arrested on animal cruelty and attempted smuggling charges (and who knows, a charge for attempting to import unlicensed animal parts may well be thrown in for good measure).