River vs Lake currents, under ice

I need a basic science lesson about water currents, please.

According to this cite, (at least some) lakes in NH do indeed have a current, which kind of surprises me given that I was apparently under the mistaken impression that only lakes that connected to other bodies of water (rivers, maybe the ocean? is there a lake version of an estuary?) had currents.

Anyway, I also had the vague impression that those lakes, the ones that fed into other bodies of water, would have a weaker sub-ice current than rivers do once both types of water had frozen over. Apparently I don’t know much about currents, so is this even close to right?

And another thing, what causes moving bodies of water to freeze over in the first place? They begin to freeze by December or January, which is before the coldest part of the year…

The currents are probably wind-driven. Assuming you have a large enough lake and enough wind/fetch, the current flows with the wind in the surface layer. When the current meets the opposite shore, the surface water “piles up,” which would set up a deep-water current in the opposite direction. If you had a really really large (say Lake Michigan large) lake this simple case might be complicated by Coriolis effects, Eckmann transport, etc. In addition, depending on the dimensions of the lake and the strength and duration of the wind you could get a seiche built up, which is basically analogous to the sloshing of water in a bathtub. In a lake it would manifest itself as a periodic (hours or days) rise and fall of the water, similar to the tides.
I think you’re right about isolated lakes having smaller currents once they’re frozen over - without wind acting on the water there really wouldn’t be much to drive currents in a lake.

The temperature of the water has to get cold enough for ice crystals to beging forming. The movement of water isn’t enough to keep the crystals from sticking together and ultimately freezing over.

Frazil ice is an interesting kind of ice that forms in moving water. Here is a real neat video.

There is water arriving through the bottom and departing that way in most lakes. Water still follows the grade of the solid layer under the soil. The soil just slows down the transfer.

Spring fed lakes will have currents around spring holes. If you happen to live next to spring fed lake you usually can tell where the spring are because the lake area over the springs will freez last.