I see at least 4 new vines you can grow easily from the later pictures, maybe more.
Just severe the sections where there is a root with an associated branch, make your cut about 2 inches from the node with the root on each side of the original vine. Plant the severed rooted vine about 4-6 inches deep or basically you want to have the first node of the remaining branch and inch or two above soil line. Then trim trim the top of the exposed vine(branch) back to a couple inches above that 1st node which you left exposed.
As far as replanting the old vine, from what I read back when I started a mini-vineyard at my old-old house, it doesn't work well. The suggestion was to cut the vine way back to nearly a stump and trim the roots way back as well and basically start over. In the first picture it looks like you have a smaller stump next to the chicken on the left. That is sort of what you want, but shorten the roots even more. You will want to get them all into the ground as soon as possible so the dormant vine starts to put new roots into the warming soil long before you get any vegetative growth. That is given your ground wont freeze solid anymore, if so you may want to refrigerate the small rooted severed pieces until then. As for cuttings, grape cuttings are easy, at least they were for me. I got around 90% success my first try.
With the cuttings and transplants, you just have to limit the new growing buds to one or two for a newly started vine because you don't have a developed root system yet, otherwise the whole deal could fail. Well maybe the old beast could handle more, but I wouldn't let it get crazy. IMO
The link Chapman provided is a good resource. I have that book, it's a good read.