Just to share a personal experience about Icelandic Air. When i was younger, my dad booked us to see a few countries in Europe and stopover in Iceland and so we flew on Icelandic Air from JFK. There was an issue with the plane. It lurched forward and came to a short stop...causing people to smash their heads in the seat in front of them. They wouldn't tell my mum what was going on and we didn't speak Icelandic, although I assumed they did speak English. My mum got up and left the plane on the tarmac. She had a bad feeling. The flight eventually got to Europe okay; but instead of landing in Belgium in the afternoon, it landed in Germany at 3am. So we chose not to use them again. However, I am sure their service has gotten better. I would be nervous though if they only considered my pet as baggage... is the area ventilated well and such?
It is $$$ to bring a pet over. My brother aked me if I wanted Hannah, my Catahoula to stay in Florida while I go explore for a few months. But Hannah is my baby; so she is coming along!