I love hiking Mt Kearsarge. My favorite hike is to drive up the auto road to the parking lot. Pick up the Lincoln Trail at the left side of the parking lot and take it to the summit, take the Barlow Trail down to the other parking lot and the Wilmot Trail back up to the summit. It can gve you a pretty good workout if you push hard enough.
Once when the road was closed, I hiked up (and down) the auto road and I can't say I'd do it again. Too much pavement. In winter, it's a snowmobile trail.
One winter, I parked at the bottom of the road to Winslow and hiked up from there. The road was packed down from snowmobiles so it made for fast going. It seemed like the trails get a lot of winter use as they were well packed out. The Kearsarge summit was my first taste on winter on an exposed summit. Temps in the single digits....windswept.....lots of ice. I had so much fun....I'll have to do it again.
In my experience, the trails are open all the time. The park facilities are not.