Call your bank ahead of time to let them know you’ll be using your ATM card out of the country. Otherwise, they will most likely flag your account, assuming there is some fraud going on.
Also, check with them about whether they have agreements with any banks in Denmark, so you could use ATMs/cashpoints without a fee.
And yes, you likely won’t be able to use your debit card at credit card swipe-points in Europe because the card readers aren’t compatible. I’m going to France soon and I’ll only be using my debit card for cash withdrawal.