There are no lockers in St. Goar. The TI office holds bags but is not open Sundays. Koblenz has lockers but you'd have to get off there, then do a round-trip to St. Goar and back to fetch bags. Your idea of getting either an English or German tour of Rheinfels, if your tour information is correct, is perhaps not achievable on this day. Contact the castle office to ask about your specific travel date, and ask them also about tours in English.
Marksburg Castle is now officially on my list for Monday after the
river cruise.
While this could work for you, it is easier to combine a visit to Rheinfels in St. Goar with a cruise than it is to include Marksburg since the majority of the cruise boats that leave Bingen terminate in St. Goar, and Rheinfels is also in St. Goar. So without luggage (and without a guided tour, if your information is correct,) a Monday visit to Rheinfels would be a slam dunk.
Marksburg: If you stop for a castle tour on the way to Mannheim on SUNDAY, this is probably the best choice. Use Koblenz station lockers, then catch the train to Braubach (just 10 minutes) and walk up. English tours are at 1 pm traditionally but you should contact the castle administration to find out for sure if this is so on your date.
https://www.marksburg.de/en/visitor-information/#/
You can take ANY Marksburg tour and follow along with a written text in English distributed at the ticket office upon request. Maybe they still use this one:
https://www.marksburg.de/assets/uploads/Marksburg__Guided_Tour_in_Englisch_2021_.pdf
Any other Castles or sights in the area that I can see on
Monday/Tuesday while using train for my mode of transportation?
Yes. The most obviously-neglected castle in this area is BURG ELTZ, which everyone seems to like. You would travel by train to Koblenz, change there to a Mosel-River line train to Hatztzenport, then catch a connecting bus to the Castle. The DB site provides all these connections. Use "Burg Eltz, Wierschem" as your destination to search for travel schedules:
https://int.bahn.de/en
Also very popular - but apparently somewhat unpopular on this forum and with Rick Steves himself - is Reichsburg Castle in Cochem. To reach Cochem: again, train to Koblenz and change to a Mosel-River line train. Because they are fairly close to each other, Burg Eltz and Cochem are often visited on the same day - so Reichsburg and Burg Eltz should also be possible in just one day, if you wish (though I've not actually done this.) Reichsburg:
https://reichsburg-cochem.de/?lang=en
https://www.german-way.com/travel-and-tourism/germany-for-tourists/castles-and-palaces/cochem-castle/
I realize you're locked into a Mannheim base. But it may be worthwhile to spring for on hotel night in this area on Monday rather than making two round-trip visits from Mannheim to see these places. At least you'd be purchasing only one round-trip train ticket instead of two.
See your pm box for an alternative outing in the Neckar River Valley - Burg Guttenberg and other castles are just up the Neckar River from Mannheim, not terribly far away.