Finnair, British Airways, Iberia, Delta and various other airlines have round-trip flights from Seattle or Portland (SEA, PDX) to Stockholm or Copenhagen (ARN, CPH) for $529-$588 with a connection. Flights February to May. Extended weekends available.
A Carry-on bag and also a personal item is allowed for free. You can avoid paying for seat selection by accepting the randomly assigned seat at the gate.