We've had reasonably deep snow, and they've been OK (as long as you have more than one so they can keep each other warm).

In terms of diseases, there are some nasty ones like red mites that can suck their blood, but disinfecting the coop takes care of that (and I've never seen any evidence of it). The only thing my chickens have really suffered from is scaley leg, but you can get ointment for that.

I also find they decide to pluck their own feathers out at odd times due to moulting or broodiness.

//edit//Also, for urban chicken keepers: http://www.omlet.co.uk/