Sound insulation - house is very noisy!

and what happens with the insulation boards with the sockets and possibly light switches?

that is not a problem. There are "extensions" for switches and sockets. You can get them at any hardware store. Cables have to be long enough, but theoretically you could even extend those.

How did the project turn out?

I wanted to point out here that the soundproof wall ideally does not have any contact with the existing wall. Leave an air gap of 2-3 cm. Otherwise, you end up building the sound bridge right away.