My experience on crashed apps on my 4G is that I get bounced off to the Homescreen. It would probably cause a total shutdown unless the apps interferes with the iOS. But it may be different from other generation models.
If the battery is functioning, you should be able to see the battery indicator falling but not when it is shorted or faulty, there may be some cells in the batteries which are depleted and not all of it.
I do not know much about battery technology so I may be wrong here. But here is my take anyways. A battery is like a long walkway when you travel on it, the shorter the distance to your destination (in this case, full discharge of the batteries). But like road, over time part of the road may turn to potholes or collapse, in which case, you would be prevented from walking over it. If you encounter that, although you still have 90% of the way to go, there is no way to crossover. I would think that in a battery, if there is some faulty cell in it, when your usage reaches to that level and encounters that faulty cell, it cannot utilize the rest of the cells beyond the faulty one and shut down.
There is a finite lifespan to the battery and 2 to 3 years is pretty much the average on how long it can last especially if you charge every 1 to 2 days.