Yes Celotex is the black sheathing material. I also heard that it could produce a smell. I am handy with electricity. What type of testing do you have in mind. I would really like to know whether it is the compressor or the fan or both. I was even thinking of calling the HVAC contractor that installed the unit (his name and number are on a sticker on the side of the unit) and see if he has any suggestions. I was thinking of the compressor blanket (this unit is a Carrier and they have an optional blanket) or maybe there is an updated fan that can be installed. From what I gather this unit was made in mid-1996 an probably installed shortly thereafter.
I have no problem using some type of foam inside the window, although it will block all light. When I heard about interior storm windows they claim it provides soundproofing. But of all the things I have seen on soundproofing they mention that mass is important, more the better. I don't know how a 1/8" Plexiglas panel will really deliver that much soundproofing. They state that given the additional dead air space and air tight fit the acrylic glazing will provide it will help with soundproofing, but again, by how much. I am not sure I want to spend $150 to find out.
I think that maybe I need to concentrate on the source itself, the outside compressor unit. The only thing is getting permission. My neighbor's house was there long before I built my house.