The felt hat is very useful for men or for women. It enables the wearer to enjoy significantly higher temperatures for significantly longer periods. The idea being to use it to cool the wearer down - that is why it is made of a pressed, felt type material.
One does not get beaten with the leaves (if there are no leaves then your host is either tight or does not like you) one is brushed with them. This is not a sado-maso exercise but rather one designed to increase blood flow to the outer part of the body, again aiding cooling and also to increase the flow of toxins from the body and out of the body in sweat. By the way sweat is very similar in composition to fresh piss, so enjoy, it is all good.
Russian banya is somewhat different to Finnish sauna in that there is more steam visible than in Finnish sauna which tends to mean that the temperature is kept lower.
The old-style black banya was not designed to get folks more dirty, by the time one gets in it should be smoke free. I am sure they are as rare in Russia now as they are in Finland. I have only ever used one once.
The icy cold dip after the sauna is, in truth, more a test for silly foreigners than it is general custom and practice. Yes, I have done it - both snow angel stylee and through the chopped ice in a lake. A cold shower is more usual and probably less likely to cause sudden death -the physical shock is intense.
Lastly, if when you go for banya (or sauna) if your compatriots are wearing bathing costume they are probably doing so out of deference to their understanding of your likely body exposure issues.