Working in the first room, playing in the second room – where do you go with the computer and what is it broadcasting?

I am just thinking how to use my computer in two rooms in the best possible way.
Work in one room, and play in the other.

Next problem:

1. The living room with LG CX9

I want to play here, so it would be enough to just stream content from Steam, GOG, etc.
Chances are from Steam, as nothing is as good as direct connectivity (neither 120 Hz 4K nor FreeSync)

2. Study with Dell U2515H

Basically you work and study, play games only in exceptional cases.

Here is my computer from the signature
AMD Ryzen 9 5900X | MSI X570 Tomahawk | AMD RX 6800 | 2 x 16 GB Trident Z (3800, CL16) | 970 Evo Plus M.2 1 TB; 850 Pro 1 TB | Mosonic TX-750 | Jonsbo UMX4 Silver

Possible solutions

1. Extend the cable from the computer (office) into the living room – not possible, at a distance of more than 10 meters
2. Stream a picture from PC (office) to living room (Steam Link – Full HD only; Steam InHome – Test it, but high latency over WiFi)
3. Put the computer in the living room, and stream the image on the screen in the office – it seems to be the best option

Assuming it is Alt 3 – how do I get the image to a screen that is not on my computer?
Also note that I must connect peripheral devices (mouse, keyboard, webcam, headphones).