what size

You will need 4 sq ft inside floor space in the coop for each chicken. 6 chickens x 4 sq ft = 24 sq ft. MINIMUM. So that means you will need a coop - just the coop part, not a run - that is at least 4 ft by 6 ft. I highly recommend walk-in height.

Any nest boxes or feeders / waterers inside will take away from the floor space needed for each chicken, so why not at least start with 4 x 8 ft?

I cannot speak to insulation (not necessary where I live, as we have mild winters with maybe and entire 7 days of low 20 degree temps), BUT I strongly suggest you put a floor in your coop! And raise it above ground a bit, at least the height of concrete blocks, perhaps two of them stacked. This will give chickens a place to go for shade in the summer. Don't use just one concrete block, as I did, as this isn't tall enough for me to reach under there to catch a frightened chick.

For a run, the necessary space is 10 st ft per chicken, so you'll need 60 sq ft, which can be accomplished in many different ways: 6 x10 in addition to the coop. That is, you can't put the coop inside 6x10 space and still count it as 60 sq ft when most of it is taken up by the coop.
If the coop is used as part of the run perimiter, you'll have to cover/enclose the space beneath the coop, too.

 

The recomendation is 4 square feet per bird, however, in a colder snowier climate I would recommend more since they will be confined to the coop more.
I have had many a coop with only dirt floor and it can work just fine. Couple things to consider. 1. make sure that where the coop is placed drains well, you do not want to place it in a place in which water will run down to the coop or you will have a muddy coop all the time. 2. make sure you do something to prevent digging predators from digging in. A heavy guage wire buried a few inches under ground and few feet out works best but I have used cinder blocks placed around the parimeter of a coop as well and had no issues.

 

General rule of thumb, and other folks will disagree, is to have 2-3 sq feet of space inside the coop per chicken. I also prefer a walk in, it helps out a lot during the winter so you can get in out of the weather to tend to the birds. If you have a dirt floor, you will want some kind of barrier outside that will keep varmits from digging into your coop. A droppings board is easier to clean. Your coop floor will stay cleaner and better smelling. I use an old dust pan to scrape my poop boards off every few days. This will help you during the winter. The chickens breath and poop produce a lot of moisture during the winter and condensation can be a problem.
As far as insulation goes, that is a hot topic. Everyone has a different opinion about it. Chickens can take cold as long as there is not a draft. Standard sizes can take more cold than the banties can. I personally spoil my birds, so my coops are insulated, but I also have banties. I don't want my babies freezing all winter. That will be up to you.