Should I keep downspout extenders installed during the winter?
Proper drainage depends on a constant gradient. These flexible pipes don't provide that. 

Those concentric corrugations and the flexibility that allows them to follow ground contour with a portion of the length dropping into a depression or having to go over an obstruction leaves a pool of water that freezes. 

Elbows and rigid downspout stock to get the roof runoff out away from the foundation works best. There are fold-up metal/vinyl 48/60" extenders available that allow for flowerbed maintenance.

If you have a lot of winter precipitation as rain, or a lot of melt-water, you need to get the runoff out and away from the house foundation, which is the purpose of a proper downspout system. Thus, extending the outflow of the downspouts is needed year around to prevent this.