yeah my buns can access the avairy through a cat flap that i have on the shed, this means the shed stays toasty and the buns can choose where they want to be.
I wouldn't although a lot depends on your rabbits. A rabbit is likely to sit out on cold wet concrete not realising it might not be good for it. Mine are all in the dry at night.
My shed gets really cold - when I work in there I need a down jacket and a heater! I'm worried about my buns, but I think they'll be ok because I have a couple of those wheat heat things I can stick in there. I'm not looking forward to trotting down there on the snowy icy mornings! At least there'll be no slugs around...
Yes mine have 24/7 access, I have one of those bendy tunnels between the shed and the aviary (all inside the aviary so perfectly safe) which means that the wind doesn't howl straight inside the shed. The shed has various boxes and hideyholes so there are lots of places to get snug and warm, but they often choose to sit outside whatever the weather (the aviary has a roof). I'm not fussed about the dangers of leaving them with run access overnight as I take the view that if it's not safe enough for night time access, then it's not safe enough for day time either.