Layers! Merino wool socks! Something like this Merino wool top ( like this ) and high vis cagoule keep me warm most of the time I find tights too hot much of the time, and have never felt the need for thermal ones Buff is good I rarely wear a hat, gloves - thin "magic" gloves do the tricky without being too hot.
Normal tights will be fine, I went running in the snow in mine. Yes to a buff. I only wore a hat on the very coldest days. I don't have any special cold weather running gear, I just wear a running t shirt with a long sleeved hi vis over the top. The only other things I have are yax trax for when it snows.
I need to get some of those magic gloves, I've got some cheap fleece ones but I get too hot half way around and end up with them tucked under my bra strap; not very comfy!
nah not waterproof, but tbh it never really bothered me getting a bit wet! i think the idea is that they do let water in, but don't hold the water, so they didn't get weighed down with it. They have fantastic grip though for the snow and ice. I have these here
Been running in Nike free runs over the summer though but will go back to those for mud and snow. You could look at waterproof socks if you are bothered by getting wet.
You can get these socks which are waterproof. Never tried them but I've heard other runners raving about them. Unless you are trail running through water or doing long distances then you probably won't need them. I went running in all that snow in January and didn't really get damp, but then I only went 5k.
Sealskinz socks are really bulky and uncomfortable. Socks with a high proportion of wool are warm when wet and much more comfy. Maybe ankle gaiters if you are running off road ? I have had goretex lined trainers which were ok, but water still gets in the top!