This does have a risk that if the backing crypto were to crash and bottom out entirely, you could see the value of the stablecoin fall below what it should be pegged to, even getting entirely liquidated if the backing crypto were to fall to zero. On the other side of the coin, if the underlying asset were to increase in value, you would not see the stablecoin’s price increase, the price would only ever go down as a result of being under collateralized.

This doesn't seem to make sense to me. In what scenarios would either of these cases be possible?

Non-Collateralized Stablecoins

I don't think they can be called stablecoins when they are unstable in nature. I've seen people call them "elastic coins", which seems a lot more fitting.