Yeah, that thing in the last pictures is called, I think, a
Hoozy Whats-its...or
Vapor Collector... one of the two. Hoses connect to it from the crankcase, breather box, and I think one other place, and it collects harmful vapors (condensation, evaporated oil...) before they make their way back into the environment and then recycles the gases back into the combustion process. There's usually a bit of a collection of nasty liquids inside the thing that needs to be cleaned out once in a while.
Can you do with out it? Yes. Will the tree hugging emissions people be happy with you? No.
Am I planning on getting rid of it with my bike? Yes.