Rosemary wants to inject herself with the shot because she knows that it will kill her, and they think it puts them to sleep so they can go "elsewhere" ; It is important because she died and it set all of the memories free to the people and this was a big problem.
In one of the memories Jonas experiences war, during this he endures true emotional pain. It is important because it is when Jonas realizes that everything wasn't always peaceful.
Jonas watches his father inject a child and kill it. It is important because, even though his father doesn't know it will kill it, he is furious at him for doing so.