I started several common hostas from seeds of ones already established in my yard. I live in central Illinois, and this year the weather forecast is for an unusually cold and long winter. The seedlings at this point are still quite small with only a few small leaves. (I have kept them mostly in the shade during our unusually hot summer.) They see really too delicate to remain outside, but I'm unsure how to keep them over the winter... any suggestions?
Those are cute! If you have a garage or basement that stays around 40 degrees F. you could store them there for the winter. Only water about once a month. Too much and they will rot. They may lose their leaves as they go dormant.
If that isn't feasible you can store them inside in a cool, dark room and water sparingly. They need their dormant time.
If you want to keep them outside, you can bury them to the rim in the ground and cover them with mulch.
https://www.gardeningknowhow.com/ornamental/foliage/hosta/hosta-winter-preparation.htm
https://www.gardeningknowhow.com/ornamental/foliage/hosta/hosta-houseplant-care.htm
Hostas in our shade garden are not thriving. After digging around a couple, we found this strange, matted rootlike material. Do you know what it is?
Those are feeder roots! That can happen when they get a little crowded, or when the soil needs a little compost to top it off.
https://www.gardeningknowhow.com/ornamental/foliage/hosta/dividing-hosta-plants.htm
https://www.gardeningknowhow.com/houseplants/hpgen/prune-roots.htm
https://www.gardeningknowhow.com/ornamental/foliage/hosta/how-to-fertilize-a-hosta-plant.htm