It's probably a good thing that you cancelled Düsseldorf, it's where I'd recommend you to go as a first time visitor to Germany. But I can give you a couple of comments about option 2.
So far, I Iike the idea of Denmark with a 4 day side-trip to Sweden.
What Swedish town/city is an easy train ride from Copenhagen (is
Stockholm my only choice?).
Stockholm is not your only choice, there are many other great options. Malmö is as mentioned the city closest to Copenhagen, but not my recommendation. There are many smaller towns that are better options for a tourist in my opinion, like Lund, Helsingborg, Ystad or Simrishamn. Or if you go a bit further, Kalmar, Varberg, Gothenburg and many more. But what I would recommend is a visit where you combine Copenhagen and Stockholm. Fly to Copenhagen, after a few days you take the train to Stockholm and then fly home from Stockholm. Or vice versa. From Newark (I had to look that up, it's helpful for those of us who don't know airport codes if you use the name/city instead) there are direct flights to both Copenhagen and Stockholm so you can do it without having to change plane somewhere. That can be a nice trip for a week, but even better if you can extend it by a day or two.
- Tour a castle and/or palace.
- Art museum (s)
- Boat ride on a local lake/river.
- Try new foods and interact with locals.
- I don't drink beer anymore, but I'd like to sample some local varieties.
You have plenty of opportunities for all of them in both Copenhagen and Stockholm. And if it doesn't have to be a lake or river, the Stockholm archipelago is a great place for a boat ride. You can take a short 30 minute trip to a nearby island just to have lunch, or a full day out in the archipelago, or anything inbetween. For castles and palaces, there are many to choose from. A personal favourite is Skokloster Palace, not that far from Stockholm, it's well worth a visit. You also got some great recommendations for Copenhagen from Cat VH. And if you want to interact with locals, Scandinavia has an advantage when it comes to language skills.
But in Europe, you have languages from all different places, etc.
Yes, and those of us that live here know that there are many languages spoken on the continent. So if you don't understand, ask if the person speaks another language. That being said, you should learn a bit of the language spoken in the area you plan to visit. You don't have to study with the aim to become fluent, but a few basic greetings and how to say thank you goes a long way.
Is it true that many European students etc. often take the train to
other places around Europe?
People of all ages do, and many also travel by other means of transport.
If I found a private room at a hostel that could change my mind.
Private rooms at hostels are common in Sweden.