Long Layover in Denmark - How it works?

Given that you are on your return trip, checking as much of your baggage to minimize carry-on load would limit your walking around town burden.

Train from CPH to the Central Train Station is fast and easy. There is a staffed widow that took chip and signature credit cards. Central Station is a short walk to Tivoli park and the National Museum. National Museum is no longer free but still worth while.

You may need some Danish Krone (DKK) but should be able to use credit cards for any substantial purchase including your train tickets.

You can easily visit the city on a 7 hour layover. It's a quick trip into town on the train - just about 20 minutes or so depending on where you are going. Copenhagen immigration is quite fast in my experience and you can be in the central city within an hour or so of landing if all goes smoothly and the lines are not too long.
Since you will need to be back at the airport at least two hours before your flight that will probably give you 4+ hours to run around and see some of the sights! I would suggest...
Rosenborg Castle (1 hour max)
A quick boat cruise through the canals from Nyhavn (1.5 hours)
Walk through Amalienborg to see the 4 royal palaces and see the changing of the guard (15 minutes max)
Walk over the Papiroen for street food (1.5 hours to walk there, eat, and walk back) - super cool place.
Sit by the water and have coffee at the beautifully designed Kvaesthusbroen pier...so relaxing. Just don't fall asleep and miss your flight!
OR have coffee in the back garden of the Denmark Museum of Design...beautiful, quite spot in the city. Their gift shop is pretty great.
There is a free walking tour from the tourist office just across from Tivoli Gardens. It takes about 60 - 90 minutes and hits the highlights of the city.
Don't waste time trying to see The Little Mermaid...so overrated.
Even though you might not have time to see all of these things, if you can get to some of them you will get a great sense of the city!
Have fun!

nothing to go through when you leave the airport to visit Copenhagen

flying out of Copenhagen there will be be security and passport checks to see if you're on a wanted list - allow 3 hours

When I flew from Copenhagen airport to US took me less than an hour from train to gate.

You can take both the regular train to the central station or the metro to a number of places in downtown Copenhagen, with the metro being more convenient for most tourist sights.
Norwegian open the check in 3 hours before departure, but that does not mean you have to be there 3 hours before, they just want you at the gate 1 hour before transatlantic flights (though in my experience you just end up waiting at the gate and no one ever check uf you are there 1 hour before.
Since you luggage is checked in, you just have to go Through Security and passport control, which should take no more than 30 minutes.

I am loving all these tips; thank you so much for the responses. I'll be having a layover with the same amount of time in late September and will definitely be using these tips.

nothing to go through when you leave the airport to visit Copenhagen

Not quite right. You will have to go through customs, but you probably won't notice.

I agree that you should skip the little mermaid :-)

Find out what you want to see and use https://www.rejseplanen.dk/webapp/index.html?language=en_EN to find the transportation options. The Metro might be a better option than the intercity train.

Check the airport map to make sure you know the route to the transportation ticket counter (it's after baggage claim). When you get there, find a transit helper (they all speak good English). Have them help you purchase the correct ticket and ensure it's validated properly (I learned the very expensive hard way that they'll nail you if you do it incorrectly). Some machines take credit cards, but you can always purchase from a live person. Some helpful hints here and a route planner here.

The airport is very efficient - even with long lines security is fast. HOWEVER, don't cut it too close especially for an outgoing international flight. Worst case scenario you have a coffee and danish at Lagkagehuset while you wait for your flight.