Yes, the Spin 5 works AWESOME with Linux Mint! I bought the I5 version at Coscto and took a chance that Linux would work and I am glad I did. Everything works perfectly. Touchscreen. touchpad, screen, keyboard backlight, etc. There isn't anything that doesn't work. The only issue I have is that if you put the laptop in tablet mode, the screen won't rotate automatically but I think that's an issue with all versions of Linux. Otherwise, everything works perfectly. The leyboard and touchpad even lock when you put it in tablet mode so you can't accidentally hit a key by mistake.
And man does it run fast with Linux!