Yesterday I installed the Linux version on Linux mint 18.3, 64bit, but the game does not start after clicking on the icon.
I started start.sh directly, but the same result.
Please, GoG, help me out.

Did you try contacting gog technical support? Did you try the fix[es] in this post, 5 posts down from yours? Do you get any messages at all when you run it from the command line? Can you try running ./LinuxLauncher directly from the game directory, if not, to see if that gives any messages?

For me on arch linux it starts but incredibly slow. Like 2 minutes before game screen kick off. Besides it has framerate issues, very long loading times even on SSD and crashes occasionally, not really appealing.