Mixer cabling?

I intend to do what is shown in the picture would it be that the people at the XBOX can't hear my game sound?

The Xbox or all newer consoles no longer have audio connections, as many things now work even more easily via HDMI.

So the question arises for me, how do you want to connect the mixer to your Xbox?

The XBOX has an SPDIF output so you could use an adapter:

https://www.amazon.de/...07MB8VCWQ/

Could the microphone be connected through the controller?

Oh yes, that's right, the old generation still has this connection, the new one is no longer the case.

It's an SPDIF output, not an input.

Then I could lead the sound to the mixer?

Yes you could do that, but then your colleagues wouldn't hear you at the Xbox Live Party.