I have to believe that the crops you name, grasses for hay, would not really be worth the effort in returns. You need quite a bit of space to get enough cut grasses for enough hay quantity to make a bale.
You would have more invested in materials for watering, dirt lifted up to the roof, fabric protection, than the price of a couple hay bales. Neither the goat or pony, will NEED a lot of hay to keep them looking good. Cheaper to purchase the hay, save yourself the work of climbing up on the barn roof to get your crops.
Also involved would be the reinforcing of the base structure to hold up the weight of dirt for growing the plants. Is your chicken coop (picturing small, 12 x 12 ft structure) heavily supported for snow on roof? Wood shingles, or even tarpaper, is not going to be able to manage the constant dampness, keep it out of the subroof sheathing. Wet boards rot and weaken. Roofs are not designed to be wet constantly. You would need to change the roof before setting up your planting system, with heavier support, waterproofing membranes to be the base of planting structures above.
Again, I can not see the cost of growing your own stuff, off setting the investment needed to do roof growing. Even purchased veggies and hay, still would be cheaper than green roof investment money.
Small, movable panels or electric fence on the ground, might let you grow grasses for loose hay. Or you could do small rotational grazing pens for the animals and birds, only allowing them on grass for short times, maybe a couple hours a day. Your small animals do NOT need grazing 24 hours a day. Both are designed to get the most nutrition from SPARSE nibbling, and STILL look very nice in body condition. Obese is not kind, can cause hoof problems.
So sorry, green roof is great idea, just not for small settings.