Really big, old clumps of peonies are always going to be a challenge to move. Some varieties make an enormous amount of roots—had one in my back garden with a single root over 4' long and it'd only been in place for around 5 years! No, I don't think the feeder roots are any more resilient in the fall but they don't need those little hair-like roots as much then and they have a better chance of growing new ones before the next growing season if they're disturbed in the fall. I've dug up a massive amount of peonies over the last few years and I think it's almost impossible to get a large one out of the ground in one piece.
As for your little shoot in the last pic it's hard to say, you'd have to remove all of the dirt and look at it more closely. I assume your peony is a lactiflora. They typically don't have adventitious roots meaning they don't usually form eyes on parts of the root further away from the crown and left over bits of roots once divided don't normally grow eyes, although I've been told it can happen. If your peony is a herbaceous hybrid it could be adventitious because of a non-lactiflora parent, those can and do grow eyes away from the crown and, when divided, pieces of broken off root can grow eyes and start new plants. Seems most of the coral herbaceous peonies share that trait. Your little shoot could also be a seedling that started really close to the parent but impossible to tell with the soil on it.
