Incorrect.
Usually widgets are doing real time updates in the background so that when you turn on the screen they are up-to-date. Think of them as programs that are constantly running in memory, not shut down when you exit. This is why Android devices don't have the battery of an iPhone. When all you have to do is display icons and no active widgets which take memory and CPU cycles, you have better battery life.
Incorrect. A vast majority of widgets, id go as far as to say 99+%, have a set interval in which they update. They "run" in the background, as in their service is active, but they are not eating CPU cycles and are pretty much just chilling in the ram so that when the user switches back to the home screen they can be displayed instantly. Maybe there are SOME widgets which constantly are running in the background, but I cannot think of a single one.