There is a high correlation between insufficient sleep and catching cold. It kind of made sense but I've looked it up and found several studies supporting this.

This winter I've tried only cold showers after running outside. Cold showers as in (sometimes) my neck hurt if I did it too long. Did not catch a cold until this spring and I think this is more correlated to insufficient rest/sleep (on top of exercising) than to water temperature.

My takeaway is enough rest is not optional and cold showers are doable even in very cold water. I have no idea how hard would be a bath in water + ice.