They do, there will be a Linux penguin on the supported OSes list if the game has native support for Linux. If the game doesn't have native support, and you buy the game on a Linux machine, it will warn you about possible incompatibility.

In any case the reality is that every game I've bought on GOG has worked pretty much perfectly on Wine, I use winetricks. The main problem with Windows games these days is the DRM which on Wine will crash. Good thing GOG games don't come with any.

It's a website; it works just fine on Linux.

If you want a fancy launcher, there's always Heroic.

Note that they recently were hiring someone to work on the Galaxy launcher for Linux