Is each “game entry” in moonlight a shared instance?

Quite the opposite, at the moment each Moonlight client will have a completely isolated session and they can play different games. This obviously defeats co-op which is something that I’d like to work on by adding proper user management and a remote UI in the next release.

How would I go about making a generic linux desktop from a given Linux distro?

In our latest images we have default support for Sway, you could easily expand that base Dockeer image with all the apps you need or make a different image with the DE that you’d like to use. The project is very open ended and only a few base components are needed in order for any image to be streamed to a remote client.