You can bike in many types of conditions with some preparation! Stay dry biking in the rain with a rain jacket, pants, and waterproof shoes. If you would have the opportunity to change at your destination, you could also bring a fresh set of clothes in your pannier. Putting fenders on your bike will prevent muddy spray from splashing on your back.
In cold weather, a buff or balaclava will keep your neck, head, and ears warm. Pogies are big mitts which go over your handlebars and keep your hands warm. You can also get a studded tire which will grip better on ice.