You are correct for the drywall covered mould, it needs to be removed completely. Now, what is behind the drywall? are the studs contaminated with mould? insulation? If in a residential area it should be removed by a licensed mould remediation technician - there is just to much liability with mould in the US - lawsuit upon lawsuit. If you start removing it you will become liable. As for the mould in the chute, don't use anything other than approved chemicals for mould remediation. NEVER use bleech, it is water based and will only contribute to the mould problem.
I rarely use chemicals but if I have to I use a mould inhibitor/killer called Benefect - great stuff and non toxic, can drink the stuff if you had to. Another good product comes from Fosters (they make paint products as well). Benefect has a scent added to it...might help in your situation.
http://www.benefect.com/ --- you would want the disinfectant