If your card has chip and pin you shouldn't have a problem using it in Denmark. I think most European Credit Cards have chip and pin now.
Yes supermarkets charge extra for using a credit card.
But doesn't your bank also charge you for taking money from ATMs?
My bank does when I take out money in another country.
So I think it'll will cost you about the same.

Cheap restaurants : Copenhagen has a lot of pizza and kebab/sharwarma places and they are cheap ( z.B. the street Nørrebrogade has a lot of cheap restaurants ).

http://www.bing.com/maps/#Y3A9NTUuNj...hycmVicm9nYWRl

I don't normally eat in the kebab places as I don't like the quality but I can strongly recommend a sharwarma place on Strøget ( the main pedestrian ).

http://www.shawarmagrillhouse.dk

If you come from the Town Square ( Rådhuspladsen ) the shop is the first sharwarma shop you see, only about 30-50 metres from the square.
Their meat is top quality and you get a lot.
39 kr for a sharwarma in pita bread and 49 kr for a sharwarma in durum roll.

The supermarked chain Føtex sells sandwiches ( 30 kr ) and salads and the like.
They also have a bakery in each shop which is a cheap bakery for Denmark.