Can we park our cars overnight at the German train stations?

I'm not sure it's uniform. The MVV (Munich metro) says on it's P&R map that the park and ride regulations posted at the station apply.

I did a few years ago when I took the train to Prague. I would check though with the station as they may all be different.

For the most part, yes.

Maybe a site search is in order here. I can't remember exact details, but a year or so ago, someone wanted to leave a car in the PnR lot in Dachau. I "thought" we found out you could, but you had to let them know at the station.

Are you talking big cities or little towns? If you want to park near the train stations in Berlin, Frankfurt, Munich, etc. you will need to pay for that privilege. They will all have parking garages or near-by parking lots.

It is a small town train station with a decent size parking lot next to it.