Cold crashing is optional, but, if you have the means it is quite helpful to clarify your beer.
A cold crash is act of dropping the beer temp to around 32F. This encourages the yeast to flocculate (fall to bottom of the fermenter.) Other things floating in the beer (proteins) also drop out easier with lower temps. Leave the beer at 32F for 2-3 days then keg or bottle.
PS. All questions are legit.
Does this still work after beer is transferred to keg, carbonated, and chilled?
Yes. It's nice to cold crash a few days before kegging to clear most yeast and big particulates before packaging, but, it's not necessary. If you skip the cold crash before kegging, the yeast and other particulates will drop to the bottom of the keg. Your first glass or two will be cloudy. After that there is no difference.
When kegging the cold crash process never stops. You beer will continue to get more and more clear. If you are patient you may allow a few weeks of cold storage (aka lagering) before drinking. In this case, your beer may be bright for the entire time it's on draft. If you are impatient, you might start with cloudy beer and then achieve brightness by the last pint served. I am somewhere in the middle.