They channel airing the show would just shift things forward or backward 5 minutes.
So something that normally starts at 12:00 would start at 12:05. The shows in the evening will start at the usual time by, over the course of the day, removing 1 or 2 adverts each advert break. The channels that keep a strict schedule just remove most of the adverts during an advert break and play the show then. They do not allocate a 30 minute slot to a short. However, if they have a 15 minute show, they would play that directly before or after the short.
Channels make a lot of money from adverts so they only remove adverts for series they expect to get a high viewing.
As for what started it, I can not say, but while I can not remember any, I am sure I have seen shorts that aren't anime, but the vast majority would probably be anime.
I think the main reason we see more shorts is because anime companies are getting bolder due to more sponsors and producing more franchise such as games and figurines etc., but I think mostly due to larger viewing numbers. 10 years ago companies might have had a hard time making profit on a short and so channels would not want to air them, but now more money means they can get better music, scripts and animation for shorts (or any other things that make them worth watching) so channels will want to air them.