Anyone tried to brush mud/snow in rear-wheel while on ride? I ride my bike in the snow. And I think my bike can be a lot cleaner if I had a snow catcher in my rear-wheel.

There is possibility because the rear wheel fork comes with extra holes in it. I think we can hook a fork there to hold a brush at the back of the tire.

I think it will keep the gear and derailleur cleaner. I guess it will also reduce the skidding.
I prefer not using anything in the snow to block the snow. You risk the snow packing in the snow blocker (e.g. fenders) and freezing, which can be dangerous. A brush against the tire (depending on the stiffness of the bristles) could also damage your tire. 

If you want to use something, a seatpost fender and a downtube mudguard are probalby the safest options. 

If you want to keep your bike clean in winter, good luck (especially if you ride on salted roads). The best course of action is to bring it in when you have time (e.g. 24 hours), clean the bike indoors, let it be completely dry then take it back out. Or, if theres a warm day, do a quick (light) wipe down and dry the bike off as you go. You could also try a stiff brush which may get rid of some of the salt.  Or just deal with a dirty bike, and clean it in the spring.