Danish lawyer here. We have rather usual defamation laws regarding libel and slander. I am not sure they differ from other European or American laws in any way that hold Danish media to higher standards. Even publishing true information can be a violation of our criminal code, for example certain information about private life or about past criminal convictions.
All in all, though, I believe we have wide freedom of press and freedom of speech not unlike what they have in the rest of Europe and in reality not that different from what they have in the US. Yes, we do have a media liability law which makes the editor (or even the journalist or the publication) liable for defamation in the content and for publishing certain information about a person's private life. The effect is that Danish publications are forced to worry about defamation against persons or corporations, or groups of people such as certain statements regarding religions, ethnicity etc or about information regarding persons' private life.
But apart from what I describe above, newspapers can publish more or less whatever they want without fearing criminal or tort liability. They mostly have to fear ridicule. A Danish editor will not be held liable if his newspaper writes wrong facts about the unemployment rate, crime, business climate, taxes or just about anything else that is in the news.
I believe the answer to why Danish media still have a decent standard despite free online news and Google and Facebook getting the ad money, is found in the subsidies. Virtually all large media in Denmark are either public entities or private entities that are heavily subsidized. By far the largest Danish media is the national radio and TV station. All the large newspapers receive huge subsidies. They simply still have budgets to pay for journalism.
Here is a list of the recipients with amounts:
https://slks.dk/fileadmin/user_upload/SLKS/Omraader/Medier/S...