They do not really go haywire, mostly stop working. There is little chance your cameras would remain operational in that weather for more than a few minutes.
There are really two things that happen in extreme cold. Starting at below 0C, most camera batteries start losing their ability to produce current. It is a slow process as the battery cools down. So you will not get an immediate failure but battery-life will drop. A few degrees below 0 and the difference will be small but by the time you reach -20C, you may only be able to take a few shots and eventually none at all. Freezeproof cameras start degrading at -10C, so remain usable until lower, around -30C and can still take tens of shots, maybe a hundred.
As the main issue is the temperature of the battery, what I do when shooting in Canadian winter is to keep a spare battery inside an inner pocket of my jacket or inside my glove to keep it warm. Then I swap the batteries each time the camera reports the battery is depleted. Cameras cannot actually tell the difference between a cold battery and a depleted battery but if it is fully charged and stops shooting after only a few shots, chances are it is just cold. Warming the battery up and putting it back inside the camera, it will not appear as depleted. In extremely cold days, I end up swapping the batteries every few shots and eventually have to wait while batteries warm up.
The second thing that happens starting at -20C is that the crystals in the LCD and EVF freeze and then the camera is no longer able to show an image. The only cameras to avoid this problem are SLRs (Digital or not) which do not need an active display for framing,
Another thing that happens is that the lubrification inside lenses loses its lubrication properties. At that point the zoom ring and focus ring becomes difficult to turn and it eventually stops being able to focus. This happens anywhere below -30C. For professional arctic expeditions, lenses are often taken apart and their lubricants replaced with something different.