The phrase about right and left hands first appeared in the New Testament of the Bible [Matthew 6:3], where Jesus said "But when thou doest alms, let not thy left hand know what thy right hand doeth". In other words: when doing good things, don't let everyone else know about it. Over the years, the phrase has tended to be used to describe an company or individual who is not well organised; who does not co-ordinate actions well. For example, the production department may have decided to stop making a type of car, but the buying department is still placing orders for the type of wheels required for that type of car. In that case, people would say that "the left hand doesn't know what the right hand is doing". Good practice would be to make sure that any major decision was widely communicated, so that "the right hand knows what the left hand is doing" [and vice versa].
To "fall between the cracks" obviously refers to a small object being lost in the small gap between floorboards. Nowadays, it is used to refer to a situation where something is missed because various people or departments assume that someone else, or another department, is responsible for that something.