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

This should give you half the answer you're looking for - you'll hear the game, your own voice, and their voice. Your friend will hear you, the game, and themselves. It's not the "perfect" answer you're looking for, but I personally do this and it works quite fine.

Simply go to your Sound Settings and configure your mic as shown:

By enabling "listen to this device", you're broadcasting your mic directly to your Stereo Mix. Likewise, your game and your friend's voice are also being broadcasted to the Stereo Mix. Open the Windows mixer and adjust the levels to ensure that all three sources can be heard, then set the Skype input to Stereo Mix to send everything to your friend.

If your soundcard has two independent headphone outputs, then it might be possible for you to not hear yourself (and your friend can't hear themselves either). Set the Skype audio output for the first headphone output, so your friend's voice will feed into that jack. Then, set the game's sound and your voice out the second jack, and route that signal (using Stereo Mix) to the Skype input. Use a 3.5mm y-connection to mix both outputs into your headphones, and everything should be all good. :D