Just a word on this first class doesn't mean the same as someone might think. First class or whatever a train operator calls it is what they define.
We had first class tickets for both Frankfurt to Hamburg, Germany and Hamburg to Copenhagen, Denmark (we had more destinations as first class but these two are pertinent to your experience). Both were 6 person cubicle style with luggage racks up top and then you'd have your personal item right by you but each one with slight variances to what the cubicle looked like. The Frankfurt to Hamburg was through train operator Deutsche Bahn but the Hamburg to Copenhagen was a different train operator (I can't remember but I believe it was a train operator for Denmark)--for this leg we were given a snack box and a drink as part of our ticket but in truth the a/c was terrible and it was hot in there, from the German and Dutch women in our cubicle this was common for that train operator.
I agree on sometimes thinking about money wasted, it just depends on weighing what that ticket gets you either less seats so less noise potentially, bigger seats, access to food/drink, etc.
We had ticket checks common-ish, it depended honestly how busy the train was and the station as well. The less busy the more likely we were to have checks. Many of our tickets were on our phones so they just scanned that or visually checked it. Interesting on the passport check I've wondered how the newly reinstated checks would be done and although your trip was a while back I could see them wanting people's passports if they were doing a country border check that was reinstated.
