Don't bend the nails down if you use the shingle method. You can actually cause a leak this way, bending them over or down can unseat the nail, thus causing water to find around the head of the nail. Don't use the shortest nail you can find, short nails can be pulled out by wind and there goes your roof. You could always use a right angle grinder to take the tips off of the nails inside but you cannot go to short. You can also sheet the inside of the coop if you have enough room.
Rolled roofing would work but I wouldn't use an adhesive for it, just nail it. Reason being is when the weather gets hot outside the asphalt in the rolled roofing heats up and can actually melt the adhesive, thus causing lifting, bubbling or curling. Constant reheating and cooling of an adhesive will cause it to fail and then you are stuck dealing with leaks or replacing the roof. Nailing is best and don't forget to use a moisture barrier under your roofing product (felt paper).