There currently is one option to travel by night train from Bad Schandau (Germany) to Copenhagen (Denmark), taking between approximately 11 and 12 hours: Namely the RJ 386. These lines are operated by Deutsche Bahn and DSB.
Deutsche Bahn (DB) exited the actual night train business in 2016. However, it operates around a dozen Intercity and Eurocity trains that run exclusively at night.
In these night trains, there are only first and second class seats. The lights stay on all the time and there is no way to recline the seats.
DSB, Denmark's national railway, runs passenger trains throughout the country and connecting services abroad.