Questions about saving potatoes for replanting

[M.SeanF.]

I just harvested my first crop of fingerlings. I hadn't planned on growing any this year, but some sprouted reds that I planted as gopher bait really thrived. So, I went down to the local nursery and bought about 2.5 pounds of assorted fingerlings. Mostly La Ratte, German Butterball, and Vienna Crescent, along with a couple of pinks and a few purple. I planted them in all the bare spots in my gopher wired raised beds just to see what would happen.

As you can see from the picture they did pretty well, considering the late planting and early heat that did them in. Since my goal was to produce "seed" potatoes for next year I consider it a success.

Here are my questions:

How do I store them until replanting next year?

I'd like to cut some of the largest tubers into pieces. How big should the pieces be? Should I do it now, or just before planting?

Will the smallest, pea-sized potatoes produce smaller plants next spring?

I've heard you shouldn't eat green potatoes, but I would assume they are fine for planting. Are they?

Thanks for any insight you can give me.

Sean