So sometimes I wish to game in the living room which is on the opposite side of the house from my office. I was hoping there was some software that I could use to stream through the local network the picture of what's happening on my desktop over to my laptop, and have control using my mouse/keyboard of my laptop. But I haven't had any success finding any software like this.

So is it possible to have my desktop computer do the work and stream to my TV or laptop on the opposite side of the house through the local network?

I believe Remote Desktop or VNC will achieve this.

The framerate is questionable, however.

why should you want to follow what's happening on your desktop, 3d applications, rendering?