I know on the weekend I can get the Schones Wochenende ticket for 42 Euro. Could I instead get the Rheinland-Pfalz ticket for 21 Euro? Is it available at ticket machines and would I be able to use that one ticket for round trip travel the same day?
Mannheim is technically in Baden-Wurttemberg (on the border) so you'd need to buy a suppliment ticket for whatever the closest town on the Rheinland side is to Mannheim, but it shouldn't be very much. You can use the Rheinland-Pflaz ticket all day long, as many trips as you want, but ONLY on regional transit, you can't use it on ICE or IC trains. Hope that helps!
No need to buy a supplemental ticket, the Rheinland-Pfalz-Ticket is valid from Ludwigshafen (in RL-P) to Mannheim Hbf (considered a border station)- also to Karlsruhe Hbf. You can read here, in German. "RP-Ticket-Strecken außerhalb von Rheinland-Pfalz:" -> RP-Ticket lines outside Rheinland-Pfalz: Ludwigshafen – Mannheim Hbf. Similarly, the Baden-Württemberg-Ticket is valid from Mannheim to Ludwigshafen.