A baby was born in heaven, usually that never happens as a pregnant women who dies work to get their baby reincarnated instead of their own life. However, Maeda's mother was selfish and reincarnated herself. Everyone is given a total of 4 times to be reincarnated. That baby lived in heaven and every 10 millennials she was able to go back to the living world for a century. That baby is Maeda Atsuko, she manages life clocks and brings those who die with her. You can say she was like a grim reaper who had wings. 

Maeda being a suspect wasn't entirely wrong. Yes she "transfered" when Nako was just diagnosed but Maeda knew exactly when Nako was going to go. (Chapter 7: Revenge)

When you saw a silhoette of Sasshi and heard animal noises, Nako called for help from Maeda. (Chapter 6: Broken)

Your time was suppose to be up, Nako just wanted you to keep her promise. She risked her last chance of reincarnation and gave that chance for you. Ami didn't have anymore chances left, your past lives used them up. 

There isn't a known reason to why Maeda stayed with the kami7 after Nako's death. Some angels say she has fell in love while other angels say she is looking for someone to take her place and help with time keeping. Or maybe it was both. Falling for Oshima Yuko and getting Matsui Ami to help her for time keeping. Maybe that's why Maeda stayed in the human realm for such a long time, at the time of the stabbing incident she has never seen any life clock go from 75 years down to a couple of weeks and shoot up to 86 years. She was told that everyone's fate was already written out, from who you marry, to who your kids are, to what your job is and how you die. So seeing this was surprising to her. 

Matsui Ami is definately someone special. You help everyone out in ways you can even begin to imagine, like a bridge that has great significance but falls if even one cable is not there.