No and yes. Wide tyres on narrow rims don't work too well because they'll roll around on the rim (for lack of a better description). Think of it this way, if you held an inflated balloon by the tied bit you'd be able to move it around quite a bit while your hand stays still, this is narrow rim/wide tyre senario (fingers close together=narrow rim). Now if you grabbed the same balloon with your hand palming the balloon (with your fingers far apart=wide rim) you'll find that the balloon won't move too much without your hand moving. There is a little more to it, like sidewall stiffness, but that's the basic way to look at it.
So you can see how this will affect things like control and the likely hood of rolling a tyre off a rim. As for dings, it's mainly down to tyre construction (single ply, dual ply, etc.), tyre pressure and material used in rims.