So it has come to my attention that (at least in the version that I use) the backgrounds aren't static. For instance, if you start a new game and go directly left, you'll get a different background every time you start a new game, even though it is supposedly the same screen. I haven't been recording this, but I thought I'd bring it up, because there might be some kind of correlation between the background used and whether or not there are gems / puffer fish on that particular screen.

I haven't noticed this happen on my version of the game, but that's very interesting you bring that up, let me know if you find anything useful!