Well sure. I mean, bundling up is possible, just ask anyone in Chicago. My hunch is that places with healthy bike infrastructure are also dense enough to make most trips short, minimizing time spent outside.

Winter in Copenhagen means –5°C, so it's not really the same. It is more humid than the continental USA.

But yes, the cycle paths are ploughed (often before the roadway). Nevertheless, bus and train use increases.

For cycling a long time in the rain, some people use overtrousers or other waterproof clothing.

Just means you can bike faster without sweating.