I had the door apart to work on my wing mirror. When I removed the speaker, I saw down the gap inside the door and there was this loose piece sat there at the bottom. Any ideas what it is and where it belongs? I would guess something related to the glass judging by the felt.

 

Correct, it's supposed to be attached to your door moulding that runs along the outside top of the door. Part number 24 in this diagram. Hopefully it'll help you figure out how to replace it.

http://www.hondaautomotiveparts.com/...G+-+PROTECTORS

 

Yup I believe they are called stabilizers, I had one sitting at the bottom of both doors, I pulled them out with one of those long pick-up tools.

They can keep falling off as they are just a friction fit onto the moulding, maybe if the glass gets ice on it the stabilizers get pulled down as the window opens. The felt material faces the glass. I put mine back in with a good dose of permatex ultrablack gasket sealer and they have not fallen back down.