Desktop sharing is simply screen mirroring. There is no controls in Chromecast. You have to control the game in your PC.
Anyway, if you play games in Android devices (with Android screen mirroring), you can control games on Android devices.
You can also get a Nexus player to play Android games with the optional game controller.
Amazon’s Fire TV (with the optional game controller) is another good option if you are a Prime member.
For PC games, you probably have to get a bigger monitor and good good speakers.