At ground level where I live has much milder winters than where you are (North Wales, UK), but I run a lot in the local mountains all through the winter where conditions can be a lot harsher and much colder. I just love running in the cold, it's all about being safe and wearing the right clothes and gear.
A buff is your friend and so versatile, it will keep your neck warm and you can pull it up over your face too if running into a cold wind.
A windproof jacket and trousers will make your life much more comfortable, windproof technology has come on leaps and bounds in the past couple of decades, I don't like tights unless it's incredible cold, my favourite legwear is my pair of Montane Featherlite pants, so comfortable and so breathable but they protect you from windchill.
Good quality waterproofs are vital in cold conditions, there are so many great running waterproofs on the market these days we're spoilt for choice, I'd steer clear of the ultralight ones in winter though and go for something a little more robust which will give a more protection in a hail storm for example, ultralightweight fabrics also go all limp and look like they've wetted out (even when they haven't) which can be quite unpleasant on a cold training run, heavier fabrics don't do this.
For snow and ice I have several pairs of shoes, my favourite are Inov8 X-Talons, Icebug Acceleritas and Walsh fell running shoes, La Sportiva Anacondas were great too but they discontinued them. For ice I just slip on a pair of Climbing Technology ice traction crampons. I don't like waterproof shoes. If I pack my crampons I always pack my ice axe too, it can be a real life saver in icy conditions.
To keep feet warm if they're going to get damp of wet can be tricky, Merino wool socks are great and still retain some thermal properties when wet but if you're feet are going to be wet for a long time and it's really cold your feet could still get very cold, neoprene socks are brilliant but they make your feet damp, sweaty and warm, which while mildly unpleasant still keeps your feet warm which is the important thing. Waterproof socks work but I find they start leaking far too soon for my liking considering how expensive they are, also they can be very thick making your shoes a very tight fit.
A decent baselayer top is the main thing though, you can get expensive merino wool ones which are very nice and great at regulating temperature, but my favourite is the good old fashioned Helly Hansen Lifa top which has been one of the most popular base layers for mountain runners across the world for decades.
I always carry several pairs of gloves but my favourite are my Inov8 Race Ultra Mitts, they're wind/waterproof and they warm up cold hands very quickly, normally though I'll just run in a pair of merino wool liner gloves. For very cold days in the mountains I'll wear Montane prism gloves and carry Extremeties Goretex Over Mitts incase it rains.