What is Denmark's main airline?

SAS, www.flysas.com

Delta is a U.S. airline. I don't think they operate their own flights to Copenhagen. Only codeshare.

SAS is the largest airline in much of Scandinavia. Norwegian Air is a discount airline that serves Scandinavia.

SAS.

SAS, "Scandinavian Air System" is the national airline of Denmark (Denmark, Sweden and Norway, actually). They've been around since the 1940s. SAS is a member of Star Alliance (along with United Airlines and 25 other partner airlines, which gives you lots of options in case your flight has issues requiring a rebooking; Star Alliance is the largest airline alliance in the world). SAS's primary hub airport is Copenhagen. A good airline to fly if you can.

I am a bit nervous about purchasing tickets on SAS for this coming June because I see they are in bankruptcy proceedings. Should I be concerned?

No.

They have been in such proceedings for a long time. It's sort of like US airlines going through government-supervised reorganization. They are the national airline of not just Denmark, but three countries. SAS is not going to disappear. They're a great airline, and I'd buy tickets on SAS without hestitation.