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Regarding the mouse frequency – it was set to 1000, I changed it to 125 and I observe the opposite effect: when I don't move the mouse, it lags, but when I move it, the picture is smooth. It doesn't depend on the graphics settings. I suspect that this is somehow related to the interaction between the mouse itself and the game; they don't seem to understand each other.

Added 5 minutes later
And yet, switching the mouse frequency to 125 made it much better – now I can at least play! Thank you, kind person, you saved a devoted fan from imminent doom ;)