anrkitec is absolutely right , there's a station connected to Trondheim Airport. It's outside, but there's a covered walkway all the way to the platform.

But I wouldn't take the train to the Trondheim city centre. Take the airportbus (Flybussen), it's faster, has a lot of stops closer to downtown Trondheim and it's cheaper. You don't have to take the train back to the airport. The train to Storlien actually doesn't stop at the airport, it stops at Hell the stop before the airport.

They might take Norwegian money in Sweden, but at a stinky rate. I recommend credit card (accepted everywhere) or change some money.