Help with setting up network with two routers + Ethernet wall outlet

The plugs on the green cables are RJ45. To wide for telephone (RJ11). So more than likely they connect to the ethernet outlets in various rooms (including his bedroom in the first pic).

Confirm this is the case. Leave all your equipment as-is. Take a spare ethernet cable, and plug it from a LAN port on your Netgear router into the ethernet port of your wall outlet. Then go downstairs to the cable box room. Plug the green ethernet cables into a computer (probably will have to be a laptop) one by one until you get a network connection. That cable goes to your bedroom. Label it or mark it with a piece of masking tape. Edit: Looks like the green cables plug into a patch panel near the bottom. Untangle them to insure they're all plugged into the patch panel. One of those patch panel ports should to your bedroom. Though I'm unsure what the blue unterminated cables up top are for.

If it works, this confirms that your in-wall ethernet is wired correctly and works. If so, then you'll be able to accomplish what you want without moving any existing equipment around. Just by adding new equipment. It's tough to tell from the picture, but it looks like you have at least 6 green ethernet cables? Buy yourself an 8-port switch. Either of these will work and are priced very competitively.

https://www.amazon.com/TP-Link-Ethernet-Replacement-Unmanaged-TL-SG108/dp/B00A121WN6
https://www.amazon.com/TRENDnet-Unmanaged-GREENnet-Switching-TEG-S80G/dp/B001QUA6RA

Go to the cable box room, and plug all the green ethernet cables into the switch. You may want to label the one cable from your bedroom, since that's the one which will be feeding Internet to the rest of the house. Eventually you'll want to label them all (I hate it when people don't label these cables), but right now it'll be less work just to plug everything in. Ideally your cable modem/main router should also be located in your cable box room, but that's a project for later if you want. Since you have a configuration which already works for you, let's not mess with it.

Once that's done, you can plug a device into any ethernet wall port in your house and get Internet. Plugging the green cables into a switch made them all "live". If you want, you can go around to the different rooms with a laptop and Ethernet cable, and test this by plugging it into each wall ethernet port.

To use the second router as a WiFi hotspot, you need to:

• 
  ■Login to your main router. Write down its IP address and subnet mask. Under its DHCP settings, set the IP address range to exclude at least 10 IP addresses. e.g. If the router is 192.168.1.1, then set DHCP to use only 192.168.1.100-255. Thus preventing 192.168.1.1-99 from being used.
  ■Power up the second router and cover up its WAN port with a piece of tape. Plug a computer into it's LAN port and access its configuration page. Set its LAN IP address to the main router's plus 1 (e.g. 192.168.1.2). Set its subnet mask to be the same as the main router's. Reboot the second router to confirm everything worked and the new settings are active.
  ■VERY IMPORTANT - disable DHCP on the second router. Once you do this, you will be unable to access the second router automatically (simply by plugging a cable into it). That's why you do this step last.
  ■Use an Ethernet cable to plug a LAN port on the second router into a wall ethernet outlet. Remember, the WAN port should be covered with tape and not used. Hopefully there's an ethernet wall outlet in your basement so you can locate it right next to your guests. Go back to the cable box room and look at the switch lights. An extra set of lights should have lit up - this is the second router. You can label that cable if you like (since you now know which room it goes to). If no extra lights lit up, then you have a connection problem somewhere.

At this point, you should be able to access the second router from anywhere on your network via its IP address - 192.168.1.2 in this example. That'll allow you to configure or modify its wireless settings. Your guest should be able to get Internet either by connecting to the second router's WiFi, or plugging in directly to one of the second router's LAN ports.