It goes from your air cleaner to you catalitic converter, however I am not sure if it circulates clean air to the cat or crappy air from the cat back into the engine either way I lost mine years ago
That's a "anti backfire" valve for the air injection system. The air injection system forced clean air into the exahust to help reduce emissions via an engine driven air pump-which may be long gone. More O2, less CO emissions.
The bell shaped piece is where a poppet valve resides. It only allows flow-one way in case of an exhaust backfire.
If you do not have to meet emissions requirements-LOSE it.