hi,
with regard to the arum lilies:
I can only tell you what I did with my arum lilies last year-my mum also did the same and hers survived. Very simple, I let them die back and left them in their pot to overwinter in the greenhouse.
the greenhouse is heated to frost free only, my instinct would be that the shed would be fine if it's going to protect them from frost.
I did the same with my agapanthus and both came up again in their own time this year. I let the pots get quite dry, but not completely dry out, began watering again in the spring. My instinct would be that having only 1 window in your shed wouldn't matter over the winter, but it may benefit to put them somewhere lighter come springtime.
Open to hearing others' views though! Let us know what you decide :)