I figured out after a day or so that I just wasn't waiting long enough for the egg to say it was warm; I was expecting an ARK-style immediate temperature. I have many many chickens now lol

you just need the fire buff in the room, contain the fire so the chicks dont derp run into the fire and cook.

Can create a room with a fire on one side, eggs on the other, and fence off the fire so they can't get in.

The Stone firepit works great, never lost a hatched egg jumping into them.

You can build a a hatchery (just a room with nothing in it). Place a stone floor block somewhere in there with a fire on top, so that it's elevated off the ground. Drop eggs and barley in there and you'll soon have so many chickens you won't be able to move around.

They'll be warm anywhere you would be warm (able to sleep). Just give the game its usual few seconds to update statuses etc after you place them, the same as you do for planted crops etc.

They dont have to be that close. Nearby should be fine. But pets iz dum, so its a good idea to have a modest barrier around your fires to stop them wandering into them.