The new Gouter hut is scheduled to close by the end of September this year. The winter room is supposed to be open after that.
As with all things related to mountain huts, it's important never to take these things for granted. Bad weather or poor route conditions can mean that a hut will close for the season earlier than planned. A year ago that happened to many huts in the Mont Blanc area, and by mid September the season was practically over.
To find out more about the situation in the mountains around Chamonix, contact the Office de Haute Montagne
shortly before heading out. They are always well informed about the current situation, and can tell you if the winter room is indeed open and what facilities it offers at the time.
On a different note, climbing Mont Blanc in mid winter would arguably be the worst time, because the days are short, temperatures are much lower, lots of snow can make it much more strenuous as well as dangerous, and you'll have to find your own way up. In other words, the odds are all stacked against you.
The normal ski route is the Grands Mulets - and you'll certainly want skis. It is served by the refuge des Grands Mulets, which opens around the end of March - before that, you can use the winter room. In fact, outside the spring season this route is normally not used anymore, because the glacier is very crevassed and when temperatures rise and the spring snow melts away, these crevasses make it too dangerous.
The ski mountaineering season only starts around the end of March, when the days are getting longer and temperatures are not quite as low anymore - but if I could chose when to go, I would prefer May. You'll need a good weather window, but while avalanches can happen at any time of the year, in winter, obviously, the risk is generally much higher. The Office de Haute Montagne can tell you all about the current avalanche risk as well, so do visit them shortly before heading out, and be ready to change your plans.