The places you are planning to visit are all very different, and most are not true castles with long, long histories. What you choose to visit might depend on what you are looking for. You may or may not want to be visiting/touring all of those places. Very briefly...
Hohenzollern and Lichtenstein are more correctly PALACES than castles, each a little less than 200 years old. Tours are available.
Neuschwanstein (built about 125 years ago) & Hohenschwangau are both PALACES with tours open to the public.
Marksburg (a Rhine Castle) and Burg (not Berg) Eltz are CASTLES, each with many centuries of history and with tours for the public.
Rheinfels (in St. Goar) is a former Rhine Castle, now a set of ruins and museum. Tours are offered.
Rothenburg is a town which many centuries ago had a castle, I believe, but no longer.
I have not visited Lichtenstein myself. All the others may be worth seeing on one level or another. I would place Marksburg at the very top of the list for its historical value.
One thing you might enjoy, especially with kids, is a castle/falconry show experience. Burg Guttenberg, on the Neckar River, seems to be roughly in your travel area and might work for you:
https://burg-guttenberg.de/