It would be very convenient to keep the fermenting feed bucket in the coop, but is it too hot? What would happen to it at a higher than room temperature? It's a long way from my house to the coop...
You can do it. It at most just speeds up the process, you just adjust to it. Once you get your batch going just keep adding, but not so much that you have a ton of excess, one batch can go for a long time, if managed properly. I just use water, no ACV or such. The 'things' you need are in the air and everywhere, they don't need help. But you can if you like… keep it covered, but not tight, it needs room to gas and expand some as it gets going. The cover is more to keep the FF clean than anything else. I often start mine in the house/kitchen and then move it out when it is ready, around 4 days for me. Your time will vary, each batch is a little different but the results mostly look the same. Just stir in the grey stuff, it is part of the process. Best to you and your birds, RJ