About the access to the open Kverkfjöll hut from the ice cap as it seemed to me when I was there last November: From the ice cap leads a easy going narrow (~200-300m wide) slope down to the hut with ice walls towering above lakes on each side with some crevasses surrounding them. These are farely obvious when travelling in good visibility but I’d recommend having a rather densely spaced gps track when closing in on the hut in case if the conditions don’t allow for a good oversight. Judging by the conditions back in November you will not need to worry about debris/ash on the way from the ice cap to the hut.
To get info on the state and availability of the hut you should contact the Icelandic Glaciological Society (Jörfí – jorfi.is – firstname.lastname@example.org). Website of the hut: http://jorfi.is/?page_id=34