OK, so a friend of mine who uses Linux (laptop with a core i5 with integrated graphics) is able to run the game fine on the highest resolution and quality settings. I'm assuming at this stage that it's a graphics driver issue and will continue to see if I can find anyone else who has experienced the same problem as you and resolved it, but so far I've not been able to find anything. I'll let you know if I discover anything useful.
In the meantime, I've tried to install and run Sunbeam on another machine with a newer nVidia card and drivers and two different linux distros, Kubuntu 17.10 and KDE Neon. On both, the game crashes on launch. I sent feedback on both. I do have a laptop with an i5 and integrated intel graphics, also running Kubuntu 17.10. I'll give it a try on that and see how it goes.
Out of interest, have you tried running any other Unity games on those machines? Do you have similar problems, or is it just with Sunbeam? I'm wondering if it could be the particular version of Unity I used or if it's a general problem. I could try rebuilding the game in a different version of Unity to see if that makes a difference. I was planning on issuing an update at some point anyway, so I'll bring that forward if we can't find any other solution.
Since my last response, I have tried to run Sunbeam on my own Linux (Kubuntu 17.10) laptop, which is also an i5 system with integrated graphics. It objected that there were a bunch of 32-bit libraries missing. After resolving that by pulling copies of those libraries from another Kubuntu 17.10 machine, it then crashed on launch with the message "process exited with code 1". I suppose it could be a video driver issue - your friend and I may not be using the same Intel driver (or i5 CPU for that matter). Another oddity I spotted in the cave *.txt file was that it listed 4 CPU cores, although my laptop's i5 is supposedly a dual core.
Huh, how weird. Well, I'm really glad you managed to resolve the problems, even if it is a mystery as to what caused them in the first place. I'm sorry you had to go through all that though. I hope you enjoy the game enough that it was worth it! At the very least you know you should be able to run other games made with Unity without the same issues.