there are plenty of riders that do that mileage and more and there are far better bikes to be commuting a distance on. The cconventional wisdom you'll get a lot of in this thread is no doubt 'for that you need a road bike (drop bars - racer type) and I'd generally agree, some people ride long distances on hybrid (barclay type but not as heavy) and I have both drop and hybrid types and happily use both whatever the weather but on windy days particularly you really do notice the positive difference being more crouched down on a drop bar bike makes - that said a lot of the time on a drop bar bike, people tend to ride with their hands on the tops of the bars or 'on the hoods' = hands wrapped around the top of the brakes
There are options within this drop bar category depending on what you intend to carry in the way of workwear, packed lunch etc and as you suggest riding through the winter will require a bike that can take a decent set of mudguards - for 13 miles each way you'll most likely be more comfortable with a rear rack and some panniers on the bike than a rucksack on you.
Be sure you consider things like this when test riding bikes and deciding what one you need.
It may be a full on road bike with a longer saddle to bars distance, very skinny hard tyres (less comfortable on bad roads & less ability to take it off tarmac) and gearing designed to go very fast on the flat but not as well up hills suits you. These do have options to winterise them and carry luggage but often at a higher cost.
A cyclocross bike may give you better options and be a more forgiving ride in terms of less crouched position, wider softer tyres whilst you are still getting into such distances - but still let you be fast and aerodynamic when you need to
You could also consider flat bar road bikes if you are worried about comfort, not bothered about battling through a windy day or have a neck/back problem that would make drop bars and aggressive bike geometry a painful problem.
There are people that would tell you the one true general purpose bike is a mountain bike and you could easily do 13 miles on a nice lightweight one with a set of smooth tyres fitted and that if you wake up and 6 inches of snow has fallen overnight you could just change to knobbly tyres and ride just as happily - again an option to think about depending on what else you might use the bike for at the weekends.
As for riding the winter - up to you, dress for the weather and there is no reason at all why you shouldn't do it, thin layers of clothes to keep you warm without getting bulky and sweaty are good. waterproof coats usually aren't very good at wicking sweat away inside and build up too much heat around your body unless you pay a lot of money and they still aren't great at everything. My preference for wearing 'cycling' gear is softshells to keep me warm and which are showerproof with a thin packaway waterproof jacket for the rare occasions when it does rain heavily.
Good quality socks & gloves to keep your fingertips and toes warm out on the ride will make your life far nicer when you've arrived and been in the warm for a few minutes or if you need to be doing shirt buttons up.
A buff to keep the chill off you face and ears is an essential piece of kit - they don't need to be super expensive, 3 for £10 in TK-Maxx rather that 1 for £15 in a mountain shop work, most of mine have been free ones given away in cycling magazines.