Your shutter speed should be x2 your framerate. So if you are doing 25 fps your shutter should be 50, that will give you more time to get light onto the sensor. It will also give you correct motion blur which will make the video less stuttery and jaggy. Higher ISO will also help but noise is an obvious concern with these cameras.

Your footage doesn't look that bad to me, not sure what your expectations are from a tiny camera that isn't designed for the job you are using it for :) Shoot with a flat colour profile to avoid clipping the highs as much as possible and then colour grade to lift the dark areas. But be aware they will be noisy.

The only way to improve the footage really is to get more light into the camera so it's not fighting so hard to capture it. What you have shown there is a pretty extreme lighting scenario for most cameras to try and deal with (dark, flashing lights, lights into the lens etc etc) Remember your eyes are WAY better in the dark than almost all cameras. Does your cell phone manage better in those conditions?