Mama frequently looks for a different nest for her brooding than the one she hatched in, I think it is instinctive for them because in the past or in the wild there wouldn't be a human cleaning up the area so the hens defense against the germs (and possibly bad eggs) that built up in the hatching nest was to leave it behind. Once again, this isn't true for all broody hens or every hatch... we have one hen who changes nesting area within 3 days and on other that stands and taps her foot waiting for us stupid humans to get her preferred area cleaned up for her to move back into!
Babies hanging out around mama rather than under her is based on average temp in the area. Over 50 degrees F I frequently see the babies around her after they are about 1 wk old during the day, just ducking into her feathers for quick warm ups and social time, though they still seem to sleep under her at night. Below 50*F it takes an extra week or so to see it. The thing is it seems that broody hen raised chicks don't seem to have trouble being happy running around in the cold, as long as mama is nearby to duck under for short warm ups.
Nest size depends on number of chicks and size of broody... once the babies are 2 or 3 days then mama needs room to stand and easily turn and preferably be able to extend her wings for a decent stretch also (at least that is the guideline I use). If she doesn't have run of the coop or outside then she needs about a 2x2 space extra (more is great though) to allow her and the little ones to get out of the nest and stretch for a bit with food and water placed somewhere in that space.
In our coop we have a 2.5'x3' broody box which is framed and has 1/2 wire mesh on two sides, it is room enough for a plastic nest box for the broody to use when she is setting and when the chicks are out and moving a lot (about day 2 or 3) we remove the nest box and just put a pile of straw in the corner for her. It has a pop door into a 3x8 area in the coop which we use for some storage and walking area for people, but is separated from the rest of the birds by framed out wall with chicken wire. We use sand for our floor, and regularly add in ashes... the flooring sand regularly gets mixed with the sand from the adult bird coop area during the weekly cleaning, so the chicks are exposed to everything the adult birds are from the start. Having a sand floor area for the broody and babies means mama hen gets to start teaching how to scratch and dust bath right from the start, even in poor weather. The wire fence between the broody and rest of the flock means the whole flock gets to see and hear the babies from the start and when they start to mingle (after about a week) there is very little drama other than a peck here and there from grumpy hens (or jealous teenagers) or attempted chick napping from our other broodies.
Everyone's set up and flock dynamics are different, so there is no 'perfect answer', just suggestions to try things out till you find what is right for your and your broody and flock. Giving mama and chicks somewhere safe for that first week can be a great help if you don't have a flock with broody and babies experience. Some folks just let nature take it's course and have great success with it, again, success will depend on the broody and the flock dynamics and coop set up.
This is also my experience and I'm in total agreement. Mine are always outside in some way with a framed broody box and run (against predators and for shelter). My "floor" of the run is the ground with a deep litter of fallen leaves and others goodies to scratch. The rest of the flock is able to see them, too, before they are let out together.
