Can someone shed some light on my predicament here (pun intended) !!

Basically I have the HX9V Sony compact camera, which when it was new, was pretty much top of the range for compacts. I use it for video which I have no issue with at all, but today during my sons birthday party we set about using it, and it was producing pics like this (below) -

Ok so the lighting was fluorescent, so everything had a greenish cast on it, and the camera really suffered with moving children, not to mention the horrible red eye.

I use DSLR cameras so I understand that my main issue here is probably light, but what I don't get is on my facebook wall I see hundreds of pictures of people in night clubs etc using their compact cameras which take perfect images, some are even nice and sharp with very high detail, what am I doing wrong or is the camera at fault!!!

Pic below, blurred out the faces!

Flash didn't come on for this one, ISO 1600 - 1/125 & F4.5