This is getting a bit off the original topic, but...

If your hands and feet are covered in sweat and frozen, likely the problem is either that your gloves don't "breathe", and/or that you sweat a lot (which is harder to solve). A lot of non-biking gloves don't let moisture out from the glove, so what you're talking about happens - you sweat, the air over the glove refreezes the sweat, you freeze. More expensive gloves and bike specific gloves use expensive material (goretex, event) that let at least some of the water vapor go through the glove material and get out, while still being windproof. If you sweat a ton you can overwhelm the material but it takes more to get there.

For shoes, cotton socks are awful if they get wet. I bought thick wool socks which have worked very well for me (if you want to know which ones I could look it up).

-----------------------

I contrast to the other posters, at 20f I've worn gloves (not mittens) made for biking but not particularly thick, and my hand stay toasty warm:

Craft Siberian Wind & Waterproof Bike Gloves
https://www.amazon.com/gp/product/B0...?ie=UTF8&psc=1

It's kind of funny actually, these gloves suck for anything that's not biking - but on the bike my hands have no problem shifting even road shifters, move around fairly freely, and are warm.

For shoes I've been wearing thick wool socks and chrome shoes which are not insulated - again it's been very warm for me. I've actually had far more trouble keeping my misection and legs warm, than I have my hands or feet.

What works for someone varies a LOT from one person to the next, especially with how much insulation a person finds they need to stay warm.