It all depends! Everyone does things differently, so what works for me may not work for anyone else.
I wean bucklings at 8 weeks because i don't want to deal with them. They either go to new homes or a dealer at that time.
If I bottle raise the does or bucklings (as a future herd sire) they stay on the bottle for 4-6 months because I have a ton of milk to use up.
If the doe or future herd sire bucklings are being dam raised, the does stay on the dam until I start to get the dams ready for fall breeding, then they are removed from their dams. Bucklings are taken away by 3 months of age so they won't breed the does.
Wethers being raised for freezer camp stay with their dams until they are shipped. I make sure they have an appointment at the slaughterhouse in early October, so I still have time to get the doe(s) in breeding shape. (if needed). Depending on when the wethers were born, that can be anywhere from 4 to 9 months.
Most of my does will not wean their kids themselves, I have to do it. They will let their adult doe kids suck, even if they, and their adult daughters, have new kids on them!
I have a dairy herd and milk the does while they have their kids on them. They give more milk than their kids can eat and if I weaned the kids early, I would drown in milk. As it is, I still have way more milk than I need when they are raising their kids, but with the kids weaned, it's awful! I make and give away tons of cheese. The dogs and chickens get it. My freezers are full of cheese and milk (for the dry period) and I still end up dumping some of it. So for me, letting the kids stay as long as possible is a good thing. It also keeps the doe kids growing fast, so they are at or over 100 pounds their first fall and can be bred if i want.
When I had Boers, they were treated the same way. I had the does milking daily also.
Where I live, the temps can get down to -50 (ok, it did only a couple of times but....) sometimes even lower with a horrible wind chill. My goats deal with it just fine. They have thick wooly coats. Dairy goats were originally mountain dwellers from the Alps, so they evolved with cold temps. As mentioned by Nancy, drafts are the killers of not only new kids, but adults as well. A draft free, dry place to get in out of the weather is all that is needed. Keep their bellies full of hay to "stoke the fire" so to speak and they will be fine. The digestive process involving hay or browse (fibrous feeds) will keep their bodies warm from the inside.
I used to breed my does to kid in January and February up here. The kids all did fine. I triend ot make sure I was there when they kidded to help dry the kids off, but once they got a belly full of colostrum, they were fine.