Actually, starting in mid to late November, the weather can turn cool enough to use the fireplace. That cool to occasional cold weather generally extends to early to mid-March. That being said, I've seen folks light the fire in the GVs as early as September and as late as April. If you need extra wood just call the front desk and they'll have a housekeeping assistant or custodian run some up to you. And when I say up, I mean all 30+ stairs. There will be firewood in a half barrel that sits outside the door to the GV already, but some people, ahem, burn right through that supply and call for more. There is already a fire making kit--which consists of starter "wood" and a propane lighter--in the room.
Some do, though not nearly as many some years ago. HHI is largely a year-round destination now, and the restaurants have mostly adjusted to that. Some of the smaller venues may close for several weeks, and even some of the more well-known eateries will close for a few weeks to refurb, repaint, and give their staff time off. However, even during the depth of winter, most restaurants are open anymore, though perhaps with more limited days and hours. Regardless, you won't go hungry on the island for want of good restaurants. As you imply, it is wise, certainly, to check on availability of a particular restaurant before you go.