"Additionally, there are cold shock proteins - wonder if cold showers can release these, and at what temperatures /durations."

I am not sure about these, but if I do a very hot shower followed by very cold long enough I do get roughly the same sauna effect. Skin is tingling and on fire. Even more so, when ice bathing.

Hot tubs are only relaxing for me. I would not want to combine them with cold water and I doubt the effect would be positive.