1) Crossing the boundary of something can mean either direction: into or out of. It depends on the surrounding text. In either case, it usually means that the person crossing the boundary is doing something unexpected and they probably shouldn't be doing it. The boundary is there for a reason, so crossing it is significant in either direction.
2) Boundary is sufficient when you are discussing one particular boundary: the edge of the football field is the boundary for where players may stay. If this particular line is crossed, you could say the player crossed the boundary.
Use boundaries for the more abstract ideas, like the morality and freedom of expression that you mention in the question. They don't have a particular, physical line for you to cross.