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As long as the roots are still pliable and not brittle, they are fine. If they are brittle, then check that the tuber is still firm. If it is still firm, then it will regrow new roots.

Yes cut them back in winter, I mulch but I am in zone 5, fertilize in spring. What I think you are seeing is the new growth. Perhaps buy a new plant if u have room, and look at the root system to become familiar the plant.

Sure, you can prune them now. More information in this article: https://www.gardeningknowhow.com/ornamental/flowers/peony/peony-pruning.htm

It will not hurt to thin out the plant, but do so now in spring. These articles will help with the powdery mildew: https://www.gardeningknowhow.com/plant-problems/disease/get-the-cure-for-powdery-mildew.htm, https://www.gardeningknowhow.com/plant-problems/disease/powdery-mildew-homemade-and-organic-remedies.htm

One of these articles should help: http://www.bachmans.com/Garden-Care/divHomePage.html?cnb=GardenCare&categoryCode=02&pageIndex=_pageIndexToken_peonyProblems, https://www.gardeningknowhow.com/ornamental/flowers/peony/why-your-peony-buds-but-never-flowers.htm

You can plant your peony in the garden at any time, as long as you're in an area zoned for peony This article has some tips for peony growing: https://www.gardeningknowhow.com/ornamental/flowers/peony/peony-care.htm
This will help you determine your hardiness zone: http://www.trebrown.com/hrdzone.html