Did you try to connect to modem directly and play cs? You have to ensure it is working OK. As to stablity issue with the router, does it happen by wire, wirelessly or both? Does it help if you open...

Since you tested powerline adapters in those two building and they can communicate with each other, I think you can purchase a powerline extender to join in existing network.

http://www.tp-link.com/en/Support/contact/?categoryid=530

You can access internet but it cannot be detected? Ensure anti-virus software and firewall are disabled. Try a different computer.

W8961ND is an ADSL modem router only, It cannot work with cable modem.

Repeater or Universal Repeater? To work with TD-W8951ND, you should use Universal Repeater mode. http://www.tp-link.com/en/article/?faqid=250