You'd have 9 days on the ground - assume you would fly into Berlin - and you're thinking 7 for the trip to Munich etc.... is that right? Berlin is expansive and interesting and typically takes 4-5 days, so you'd likely be shorting it unless you split your time more evenly between the two areas. Fly back home out of Munich so that you maximize your time on the ground.
Yes, Germany has trains; they go nearly everywhere, and of course between Berlin and Munich. Railway map:
https://mdeen.home.xs4all.nl/temp/dbkaart.png
If you visit the Netherlands, you might also spend a couple of days in Germany's real "castle country", the Rhine Valley (south of Cologne.) These castles have medieval roots - some are intact and look much as they did 800+ years ago:
https://www.romantic-germany.info/things-to-do/culture-castles/castles-palaces/
The region's river valleys tend to make for pleasant bike rides. You can hike between castles and towns on the Rhine Castle trail:
https://www.romantic-germany.info/index.php?id=2940
Of course the Netherlands is nice for biking too.
Neuschwanstein was built near the dawn of the 20th century; a face lift made it LOOK like a castle, but it's smoke and mirrors - not a real castle.
Will you be taking your own bikes to Germany? By train + bus it takes nearly 2.5 hours round trip to reach N'stein from Munich. Seems to me like it would be quite a bike ride, especially with baggage.