Connected to internet and can browsing but can't play some games
Looking at your connection it does not shows full WiFi signal strength?! Why is that?, Try getting closer to the source.

After reading the game requirement, you would be connecting to they game server.

So most probably the problem is not your Internet setting but they server, since your friends Android works on your network.

Try Pinging they server to see if that works.

Reset your DNS settings to standard, thus remove any other settings but your router. The DNS should be issued by your ISP provider.

Go to Starbucks and try it there.

If not, try using a proxy server.