The first part is memorable. The 2nd part takes some concentration. The 3rd part is also memorable but jumps to the 5th position, but you're probably right. It could be an octave lower.
So......out of 'Nothing" comes 5-step waltzes!
Here's one I was looking at. Unfortunately Ira Ford doesn't provide any History in his book so I have no way to date it before 1940. I put in a version in E in case any intrepid banjoist wish to try it. I have a metronome app on my phone that I can set to 5/4 and that is a big help when trying to make sense of something like this. F is the original key.
Well, that's obviously the same one I posted but altered, somewhat and condensed. I got it from "The Home Circle Collection of Piano-Forte Music", 1859.
Here's an additional monkey wrench. I started going through "Home Circle" for more 5/4 waltzes and this 2/4 waltz came into view. As I started to play around on it, I immediately recognized it as one that I have a recording of (somewhere). I suppose the high parts could be played an octave lower but I'm trying to get comfortable in the 3rd position and this one isn't as difficult as some.
Oh, I didn't include the 4th part.