Cold crashing is a term typically used to describe the process of taking the temperature of beer in the fermenter down to near-freezing in order to precipitate yeast, break materials, what have you. The end goal is clearer beer.
It is also used to describe doing the same thing in other vessels - yeast starters, for instance.
Cold crashing is certainly not necessary, and whether or not you want to do it is up to you. I personally do not cold crash as I find letting the beer sit in the fermenter for 4 weeks provides beer that is very, very clear (except for things like hefeweizens, which should be cloudy).
Pitching cold yeast in to wort does not require any stirring afterwards. They'll wake up on their own.