How do I voice-chat with my friend on PC, sharing my screen and game audio without mixing in their voice?

I ended up using Voicemeeter. It has a pretty easily understandable user interface. Didn't work at first, but I managed to pick the right settings and after a few system restarts it's all working just about as I drew on my diagram in the question. The app needs to be running for audio routing to work and you can close it to return to normal without having to restart if you configure it correctly. Did I mention there are no sound artifacts or delays introduced by this solution? :)

As of the time of this answer, the new version of Skype for Windows 10 allows you to simply Share your screen, along with system audio, sans Skype. Thanks, Microsoft! :D