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As for what the dust does, can you trace it's path back to anywhere at all?....a vent opening (new or old). I'd be doing my best to remove/move the duct before I dry walled over it, turn on your furnace and see if any air is pushed through the duct, gets warm etc.... Is it near a kitchen, sometimes I have seen weird ducting for exhaust fans.
Yes you could glue the drywall to this duct (it's not the normal practice), make sure you have never seen any condensation on it first though, if it has had condensation on it in the past then you may run into problems. I'd use construction strength builder adhesive to do the job.
As for the corners you can buy at H.D and Lowe's (Lowe's sell 1/4" drywall, I saw it today) inside and outside corners that don't need nails, they look like pre molded tape you simply apply mud to the corners of the drywall the apply the "quick corner" as you would normally lay tape, let it dry come back and lay on the second and third coats and your done....no nails, I've used them many times in situations similar to what you describe.
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