Train from Paris to Denmark

No. Its 13 to 15 hours depending on where in Denmark you are going. That puts it in flying range for me.

I have.
It was a last minute trip in peak season, and flying would be 2x-3x more expensive than buying an Interrail pass and take the train. We saved around 500 USD per person by taking the train instead of flying - that made it a no-brainer for us.
On this route, flying is always much faster and 95% of the time it's cheaper than taking the train. Most of the time, I would probably fly too.

The train journey took around 13 hours. We left Copenhagen in the morning, changed in Hamburg and Essen before arriving in Paris in the evening. The long distance trains are comfortable as always, but it's a very long day on the rails.
If you take the train, you could consider staying a night somewhere between Paris and Copenhagen. On our way back from Paris, we spent the night in Cologne and got to explore that city for a couple of hours.

I realize that such a long train journey isn't for everyone. I would strongly advise against it unless you're like me and can sit back with a good book (or podcast or whatever) and really appreciate the changing landscape, architecture, climate and languages as you're travelling through Europe. If you are like me, I would still think twice before doing it. The trip itself is not exhausting or stressful, but you're spending a lot of time by getting from A to B.

If there was a night train between Copenhagen and Paris, that could be an obvious choice. I'm told there used to be one, but that service was discontinued about 20-30 years ago (shout out to the EU for effectively wiping out most night trains).

There was indeed a night train from Copenhagen to Paris, also in the past there were direct night trains from Copenhagen to Stockholm, Amsterdam, Basel, Munich, Prague and even a direct coach to Moscow once a week, attached to the Berlin train.