Frankfurt to Munich - Driving or trains?

Google is useless for trains. What you need is a map which actually marks rail lines and stations, not roads.
You can look up train times and fares on the DB (German Railways) site: https://www.bahn.com/en/view/index.shtml
You need to enter town names, not names of castles!

The Rhine Gorge is west of Frankfurt, so you would go there, and then come back through Frankfurt. There are rail lines along both sides of the Rhine Gorge, so access is easy.
Alternatively, since you are coming from Amsterdam, and trains from Amsterdam to Frankfurt call at Cologne (Köln), which is the opposite end of the Rhine Gorge, you could do the Gorge before Frankfurt.
Or you could get a car just for this area.

Burg Eltz is not on the Rhine, it is on the Mosel further west. The nearest station is "Moselkern" (look on a map), but I think you need a car to visit it. There are plenty of other castles on the Rhine, see here: https://upload.wikimedia.org/wikipedia/commons/c/ce/Karte_Mittelrhein.png

Frankfurt am Main - Rothenburg ob der Tauber - Munich are long runs best suited to trains. BTW, use the full name ("Rothenburg ob der Tauber" and "Frankfurt (Main)") as there is more than one Rothenburg and more than one Frankfurt in Germany.
Dachau is a suburb of Munich, and easily done by local train and city bus.
Neuschwanstein Castle etc. is south of Munich, near the Austrian border (nearest town Füssen). You can do this as a day trip, or stay overnight.

This is the official website for Burg Eltz: http://www.burg-eltz.de/en/
It appears they run a shuttle bus from the station, but only on weekends: http://www.burg-eltz.de/en/planning-your-trip-to-eltz-castle/arriving-by-arriving-by/arriving-by-public-transport.html

As to Burg Eltz, you can see from Chris's link. The station you get the weekend bus is not the same as the one to hike from (Moselkern). Only do the hike on a nice day, as it is through the forest part way and up hill on an unpaved trail. About a 1 hour hike from the station.

Welcome to the forum, jmshebest.

Eltz - Rhine - Bacharach - Rothenburg - Munich - Neuschwanstein - Dachau

This well-trodden tourist route is no problem by train (plus the walk or the Burgenbus to Eltz.) But before you plan out your route and figure out how to get where, I will just urge you to think some more about the destinations you have in mind and the distances you are covering in your 5-6 days. I respect much of what Rick Steves has penned over the years - I've been a fan of his for decades - but not his itinerary planning guidelines for Germany, which it seems you are pursuing, and which you have augmented with Dachau:

- 3 days: Munich, Bavarian castles
- 5 days, add: Rhine Valley, Rothenburg

IMHO and experience, this will take you on a semi-frenzied pace on a trip that emphasizes the most touristy of touristy places in Germany... no "back doors" here at all. You will likely see more of Germany if you limit your mileage and spend more time in fewer places. It would be wise to consult some additional sources - there are many good ones - and maybe to seek some itinerary advice here on this forum as well.

Day 1 in Germany starts in A'dam, right? After your 5-6 days in Germany, will you be flying home or going elsewhere in Europe?

Use this webpage for the schedule to Burg Eltz. Chris's link will get you here eventually; why not start here? You can put in the name of the castle. If you put Burg Eltz in the "To:" field, it will give you some options, including "Burg Eltz, Wierschem" and "Wierschem, Burg Eltz (Tourismus)". The first selection is the bus stop above Burg Eltz. The second even shows you a 12 minute walk from the bus stop to the castle.

The train from Koblenz stops at Hatzenport; the bus takes you from there to the bus stop above the castle.

Make sure you use a Sat. or Sun. in May; if you use a weekend day in December, you won't get anything. The bus only runs in the warmer months.

If you are starting your planning based on Rick's Germany guidebook, it includes detailed information for reaching Burg Eltz, with reference to the same links above.

Dachau and Neuschwanstein are different directions from Munich by public transport. If you rented a car, then you could skip some train routes that require multiple connections and you could use it for the Bavarian castle region before turning it in. Since Dachau is so easy by S-Bahn from Munich, your visit need not be during the same period that you have the car, although you're very flexible while you do have the car.

Train would be do-able. As others have said, it is a rushed itinerary. Put in times on the dbahn site to get an idea of the time involved. If you feel you must stick to all these places, I would get a car.