How far are you willing/wanting to go? If you take the Straßenbahn 4 or 9 to the end, you will be in Bad Dürkheim. From there you can find a lot of really nice hikes into or along the Pfalz.
In the other direction, you can take the Straßenbahn 5 to Weinheim or Schriesheim. From Weinheim I found this hike past two castles and some nice forests quite nice - I've done it a few times with visitors. From Schriesheim I like this hike by a castle ruin and over a mountain quarry with great views.
If you have access to a car and can go further away from public transport, I can suggest some other nice hikes in the Pfalz or Odenwald.
For regional transit you can also use moovit (has more regions and a few more features but has ads) or Transportr (free and open source, but has less regions for now. However for Mannheim that doesn't matter as it does have VRN).
As far as I know, both of them get their data from local transit authorities, so they are as reliable as those are, are available for cities all over the world, and have real-time updates where those exist (VRN does have them).
this one on Airbnb looks pretty decent. https://www.airbnb.com/rooms/10354089?checkin=02%2F19%2F2016&checkout=02%2F29%2F2016&s=Lw2Ji_NY
other than that there looks to be many other options for under 30 euros / night, which really is not bad. i don't think you will find any better deal than that, unless someone here offers a room for free.
the youth hostel (jugendherberge) in town will be 26.20/night (if under 27yo) or 32.20 (if 27 or older) with breakfast included but in a multi-bed dorm
Would it have good enough coverage in Feudenheim near Mannheim? I checked coverage but the street flat is located at is not covered by either 4G or 5G it seems: https://www.nperf.com/en/map/DE/-/169022.Vodafone-Mobile/signal/?ll=49.48262442823028&lg=8.534746170043947&zoom=14
https://www.amazon.de/Omron-HBF-511B-E-Ganzk%C3%B6rperanalyse-Waage-BF511-blau/dp/B0033AGBW0
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Ich kann die hier empfehlen - du hast einen abnehmbaren Handteil, dadurch ist die Messung genauer.