Most of the functionality of the getting images from the operating system onto your monitor is incorporated into the video card driver. read more
While they don't actually use a driver many people believe the info file that contains the monitors supported resolutions and refresh rates to be drivers. It sounds as if it might be heat related. More pixels of the larger monitors could be causing the card to overheat because its working much harder than on the Dell. read more
it depends on the monitor - any features beyond "display this image" will likely need drivers, it's possible to adjust settings on the monitor via the control panel, and that sort of thing with some monitors, others will return information (like the orientation of the monitor. read more