I love the fact, that the chat fades out. It makes the view clear and enyojable
But after the chat fades out there is still the Input-Field and Social-Button left over.
It would be great if these also would fade out.

Social button fading is coming in TWW

As for the "input-field", you're using the "IM" chat style, it's a Blizz "Social" setting.

That edit box properly fades out when you use the "Classic" style.