I have no experience with raising chicks in cold weather (I live in Kenya and we don't do winters here - thank goodness) so i will let others with experience on that subject help. Yes, leave them with the mother, but be aware that she needs to leave the coop to poop, stretch her legs a few times a day. A way of getting around that is to build an "extension" to the coop that you can connect to the mini-coop - that way both the mother and chicks have more space. Just took this pics to give you an idea of what i have:
The part on the right is the mini-coop that i can move around (as well as put in the main coop on a night time) and the other part is actually an isolation coop, laid on its side. Here are the two once disconnected.
The duration that you should keep them in the mini-coop depends on a number of variables, including temperatures and predators. If its cold, then the chicks should not really be outside for too long until they are 4 weeks old (thats about the time when they should be feathered and able to regulate their temperature better). As i say, here in Kenya we don't have that problem, so i let mum take her chicks out on day 2 (but supervised to keep hawks away). They are now 13 days old, and i let them free range as they wish - the mum always brings them back to the mini-coop at 5pm sharp every evening! I am likely to lose a couple of the chicks to predators, but its something i have reconciled (doesn't stop me shouting and swearing for a few minutes when it does happen though! ).
Things may be a little different re: temperatures for you, so i would search for threads on that subject matter for your area. You could also search for threads from BYC members in your area as they are more likely to give you the specific advice that would be best for you.