Thanks! Ok, here is the scoop for NYE with kids in Lincoln.
South Mountain - very nice resort right in town. They have their original building and they put on a new wing to that about 3 years ago. Most of their 1BR units are the same (and you will be happy with them). There are I think 4 1BR units that sleep 6 - these are rare and consist of a room with a day bed in it. (it really is like a 2nd bedroom). If you reservation says sleeps 6 you may have been lucky to get. I don't believe you can request it if you only have sleeps 4 on your reservation. If I were to stay there, the only requests I would make is a unit with an oven (not all units have) and a balcony. I just like having a balcony - not really a need in the winter though.
Area - the resort is right in town. You can walk or take a 2 minute car ride to restaurants, grocery, movie theatre and even a small live theatre at Jeans Playhouse (sometimes during school breaks they have family friendly things like magicians).
Skiing: Loon is right down the street, Cannon and Waterville are about 15 minutes away and Brettonwoods is about 30. We always drive to Brettonwoods. Less crowed, more terrain, better snow. Also, they have an incredible Jr. Pass program. Basically rent your kids equipment for the year at a participating ski shop and get a season pass for your kids that offers perks like $15 off adult tickets. Last year we paid $199 for brand new equipment lease for the season and she skied free all year long!
https://www.brettonwoods.com/juniorpass
NYE: Not too much in the immediate area for celebrations with kids. Waterville Valley a couple exits away usually has fireworks, and Riverwalk resort (right next door to SM) has had them too in the past. I would make the trip to Santa's Village for their NYE celebration, that it what we have done the last couple years and the kids will LOVE it. Tickets usually sell out before the event, but well worth it. Yes, it can be cold - but if you got all your ski gear and some hand/feet warmers you won't even notice. Just about all their rides are open, all food is included and they have a long firework display. This I would say is a must do.
https://www.santasvillage.com/park-info-tickets/special-events/
If the weather is cold enough the Ice Castles may be open by then.
https://icecastles.com/lincoln/
Snowmobiling: We take tours with:
http://sledventures.net/ Our DD loves it - not sure if they have an age restriction for the passenger. They usually have awesome deals on groupon.
Ice Skating: Riverwalk next door offers and I think Loon Mtn still does.
Sledding: The Kanc Rec Center in Lincoln offers.
If the kids need to burn off some steam - this place offers drop in play time a few times a week:
https://www.livealittlefitness.com/kids-programs
Ok, that should give you a good start. Let me know any specific questions you may have, happy to help.