Well I'm unfamiliar with the temperature that they are brooded at. I also have turkeys and brooding them is different than chickens.
I'll give my educated guesses. The shed should be fine as long as the brooder holds the temperature you want it to. I would raise it off the floor to avoid...
When are you brooding and how many? Chicks will quickly out grow those after two weeks. I brood in a homemade brooder box with a suspected heat lamp. I wait until May or June so temperatures aren't that big of a worry. I brood out in a shed, I don't think the shed is a problem. The brooder might...
You can use the crate, you just need to surround it so there's no drafts, or push it against the wall and cover two sides a bit so they have a cozy place without drafts. I would move them to the coop when you are ready and feel comfortable doing so. I wouldn't keep them in tubs either, they...
I would also keep them in the garage but you need something with solid sides like donrae said, I would cover the top with a screen or wire to keep the cat out, but you may find him sitting under the heat lamp, and depending on how cold it is you may have to block off part of the top to hold...