Renting in Copenhagen - advice?

One thing you can do to ensure that a contract is valid and fair on both sides is to make sure that the contract is written on one of these contract forms:

1. Typeformular A, 9. udgave

2. Typeformular B 1998

They are commonly used by professionals and has been issued by the danish authorities. You can find a english translation of the first contract form here: Form A, 9th Edition. It's important to read § 11/section 11 carefully if the first form is used.