I'd like to go skating on ice on frozen lakes and walking in the snow with snowshoes. Is February and Are a good time and place to do so?
February tends to be a month with snow to spare.
Both your activites can be done both in Are and the nearby arger town of Ostersund.
Both easy to get to from London. Fly to Stockholm and catch either a connection flight to Are Ostersund Airport (located just outside Ostersund) - flysas.com for prices and schedule. For connections to Are either take the bus into Ostersund and get a train or coach from there or use the Airport Transfer service - flygtaxi.se/products/transfer/are/ostersund/.
You can also fly to Trondheim (in Norway) with Norwegian.com, but their schedule isn't very good as the London-flights land in Trondheim at 11:30pm, so you'd need to stay a night in a nearby hotel (f.x. the Rica Hell Hotel - something to brag about? Stayed a night in a hotel in Hell?). And then catch the morning train (8:20am) to either Ostersund (3h 20m) or Are (2h).
As previous person said, ICe-skating on the lake is larger and better in Östersund so you can fly there and then first do the ice-skating before you head out to use snow-shoes. There are some nice places around Vålådalen for using Snow-shoes.
February might be cold so if you can I would suggest the beginning of March, the Ice-track Medvinden on the lake close to Östersund is open then.
Åre is very busy during winter breaks, from mid-February to late March. Beds are usually fully booked.
The main attractions of Åre are the mountain with its view and ski slopes, and its wild nightlife. If these are not necessary for your trip, Östersund would be a better choice. Or maybe some place in Dalarna or Lappland?