Yes, that is my experience with RIR, going back to childhood. I have one left in my current flock and he is my mean roo.
I gather that nutritional deficiency and shortage of space are probably the two most common reasons for feather picking, of causes that we can do something about, at least. That is a minimal amount of space for 10 birds, and not enough for 18. Enlarging the area might or might not help the feather picking. Sometimes it indicates a protein deficiency. You could offer a protein treat every day (scratch is low in protein, so dropping that may help, but layer isn't real high either. Maybe a small bag of game bird feed for a treat, or meat scraps.
On the other hand, I have 4 hens who are true free range, wander over a couple of acres and could go further, yet one keeps a bare spot on her back from the others picking. It's not poor nutrition. I think sometimes they just get in these habits. (The rest are fenced, but both the coop and yard are quite oversized.)
Egg eating is very hard to stop; many on here just cull the birds. You only need one nest to 4 hens, so 3 would be plenty for you. You could replace your nests with 3 rollaway nests. You could try hanging a sort of curtain over the nest box, like a scrap of tshirt most of the way over the opening, as they don't seem to eat eggs in the dark. You could try putting golf balls or other fake eggs in there to frustrate their attempt to eat them.
If you do not routinely treat for worms and for lice/mites, these would also be things to consider. As parasites, any of these in effect steal the feed from the bird.
On the FAQ page there are links to some good threads on feather picking. And a quick search will give you lots of results on worms and lice/mites.
Good luck.