How to voice-chat with the friend on PC, sharing the screen and game audio without mixing in their voice

I consider myself a tech-savvy person, but this audio-related problem I don't know how to solve.

Imagine the following situation: You're chatting with a friend using Skype to show them what's on your game screen. They can see the picture, hear you, but can't hear the sound from your game.

In Skype's audio settings, you can set input (what your friend hears) to your PC's sound mixer output. They will then not be able to hear you, but they'll hear both the game and their own voice.

I have tried using Teamviewer and streaming applications for this purpose, but my friend hears himself in addition to terrible lags and latency.

Using separate devices is also pretty bothersome:

• Show game and send it's audio via Skype (or any other app that can share screen) on PC;
• Chat with friend on another device (friend can use both apps on the same PC) while combining your own audio by using a 2-input → 1 output 3.5mm audio adapter (risky business on its own);
• Alternatively, use another voice chat app on the same PC and some configuration of virtual audio-mixers so that you can hear both the game and your friend.

There's gotta be a better way!

How do I properly configure my PC software so that I can:

• Voice-chat with my friend;
• Share my game's video and audio with them;
• They won't hear themselves speaking?

Update: Today I tried a bunch of third-party apps like Virtual Audio Cable and such. They all introduce horrible noise due to how much CPU power it needs to perform fast without errors. Setting the latency to higher values fixes the noise, but duh, the latency becomes huge (~4 seconds). Either I'm not configuring any of it right, or it does require superior processing powers to perform flawlessly.

Note: I sometimes play some tunes on a midi keyboard in FL studio and I know how well my PC handles a bunch of instruments with the flawless sound at very low latency (<15ms) thanks to ASIO, so my computer is not very slow and that software shouldn't be underperforming so badly.

Update 2: ASIO and multiple sound outputs got me thinkin. I have a pretty cheap Soundblaster audio card, which has multiple input and output channels that I'm not using. Maybe I could manipulate the sound flow with an audio program? Something about sending the game's output to an unused "line out" device, grabbing it with an app and sending both to me and to an unused "line in", mix in the microphone input, which Skype would then pick up.

Update 3: After some experiments with rerouting inputs and outputs via ASIO, it turned out that Skype refuses to use any audio devices involved in such tampering.