Diagnostics are not too difficult. Plug the camera into a different monitor and see if things improve; if they do, it's the monitor, if they don't, it's not! Your problems could also be caused by the cable between the camera and monitor, so try the existing camera and monitor with a different cable to see if things improve. Our system consists of a fairly decent CCTV camera running through a video splitter/booster and then to several TVs around the building (we have a video patch panel in the control room and breakout boxes through the building); none of the TVs is a top-end model but they serve purpose pretty well. We could theoretically record shows through it - just plug one of the splitter outputs into a recorder, but we never do, mostly because the camera doesn't have a microphone and so it would be a silent recording! It doesn't have night vision, but then due to the nature of our shows, there's almost never a time when I need it. I have worked with systems where the camera flicks to infra-red below a certain threshold, and provided you have a decent IR source then it works well. I've no idea what the angle of the camera is - but it's pretty wide, which is necessary due to the shape of our space. It auto-focuses, which 95% of the time is great, but sometimes it does go out of focus with a new lighting state - but the next state change beyond that normally brings it back into focus. Our prompt desk is in a corner where you have no unobstructed view of the stage, so the monitor is very important, and it serves us pretty well.
Hope that helps you some!