Cute coop! It totally looks like a blue chicken and the front legs are too cute!
You don't really say what kind of winter you'll be dealing with...so I'll just use my climate for example.
One winter we got 24 inches of snow in one night and then two days later another 18 on top of that. During that time I had 4 standard size hens (large fowl).
What I will tell you is chickens that have their feathers in can withstand cold weather fine. Some breeds are hardier than others...but it isn't so much the cold air that would make them sick.
What makes them sick is a draft blowing directly on them, them being wet, dampness in the coop, mold or bacteria in the coop.
Your coop has great ventilation for the three seasons with that whole top being open (summer heat can be worse for chickens than cold), However..it looks like your roost is in direct line with that opening up top...which means in the winter the wind blowing might be blowing directly on them. It's tricky because you will still need the ventilation to let the moisture from their breath and poo get out. What I would suggest is see where the birds are when they are on the roost...if the wind would be blowing on them then maybe consider a clear plastic tarp (shower curtain..ect) that you can drape over the entire coop. What that would do is act as a wind block but still allow for pockets of air to move since it would be hanging off the roof..not flat against the walls. It being clear would allow the sun to get in until you can let them out.
As for the floor...cold air drops...so unless it was a super windy day/night I don't think you'd have too much issue with the bottom door. So then it just becomes your choice if you want to let it stay open for them. During those days that the snow was so high I couldn't even get to my chickens to let them out...so it was good I only had 4 (my coop can hold 8 tight) So you keeping the floor open might give them some place to hang out if you can't get right to them.
If you're not dealing with snow...just cold air...then leaving it open should be totally fine.
Just a heads up..if there isn't enough ventilation your hens could get frostbite. Frostbite on the feet will look like black scales and on their combs or waddles it will look like black dots.
Hope this helps...I'm sure others will offer suggestions as well.