— Is the glacier still moving in the winter?
Yes. This photo shows the front of the glacier, and a pool of water sits on top of the ice in front of it, despite the fact that it was not much above 0 F. That's because the glacier is pushing forward, deforming the lake ice (in this case pushing it downward.) While we were there, there was a near-constant sound of popping as the brittle ice made way for the glacier.
However, the speed of glaciers does vary - it tends to be highest when there's a lot of water in the holes and cracks in the glacier. This water essentially helps lift the glacier, creating less friction at its base. So in cold dry conditions in the winter, the glacier does move more slowly.