Electrical – What kind of sound input socket is this

This is a GE device, though I think a Leutron or Leviton also make a pretty similar unit. Here's a link to it on Home Depot's website.

My experience with these units is that they fail very rapidly - I went through 3 of them in a year before I switched brands. You can see the comments on Home Depot echo this. They also can fail closed so that with a dead switch, everything is still powered and you have no way of turning it off - quite annoying if you ask me!

I would recommend that you replace it with another brand. They are not serviceable.

There are a million options out there. The main thing that you need in this setup is finding a receiver that has really good output control - in that you can pick exactly which outputs you want the sound to go to and be able to save those settings... So if you want to go to all rooms you would have an all-room criteria saved, and so on. You need the receiver and the monitor/module that controls it - usually sold together. Everything else is all about tastes and how much money you have to spend, and how many controllers you want - you could station wireless controllers around your house.

Don't want to get too off-topic for DIY... You need to be looking at multi-room or multi-zone receivers. Here is an example. I am not telling you this is the exact fit for you. You have a lot of researching and figuring out your requirements. There are receivers that have monitors that you could station in every room and every room could hear its own thing... lots of $$$. I would say you need at least 2 zones - living room and bedrooms... Also can you control the sound level in the bedrooms do they just have the speakers wired to the living room? If you want different sound levels each room will be its own zone.

Also 5.1 isn't a must. There are tons of surround options from 3 to 9+ speakers. 5.1 was the standard a while back and is fine. Any multi-room/zone receiver will give you surround abilities (most high end ones do 7.1 or 7.2) in your main living space.