I’m not sure I follow. You can terminate your cables into wall mounted sockets just like your electrics. What’s the issue with this?

No, because a switch can’t route traffic from one point to another. Specifically, it can’t do NAT. Without NAT you can only have 1 device connected to the internet.

If you move it and it changes behaviour, that’s some sort of bad mechanical contact. Cabling, socket.

Are you sure you know all the rooms in your home? Is the house bigger on the outside than you can account for on the inside?

Are you behind cgnat?

Can you post a screenshot of everything? Can’t tell what the link is. Your PC to the router? Your router to the WAN?

Use vlans to create a dmz you can place a server only for public Internet facing stuff.