Ok, stay with me, this may be a bit of a ramble- but I really am in need of some help and advice.
I have 1 Roo and 12 hens, all about 11 months old, and I have had them since they were day-olds from the feed store. (I also have 2 of my own barnyard mix hatched chicks, but they aren't integrated yet, so no issues there)
Issue #1- I have at least one egg eater. My question is what to do about it or how to find out who it is so I can take care of it. I still get a nice number of eggs, so I'm not too worried about it, but I would like to stop it before it gets worse or others catch on and start doing it as well.
Issue #2- and my biggest concern- All but 4 of my hens are being pecked/feathers picked. Even my roo. I only have 4 that look nice, all of the others have bare backs and some (like the roo) have bare spots near the neck as well. Sometimes they have scabbed spots and sometimes it's just red and bare. I tried what they feed store gave me- Rooster Booster- that was a purple cream-type stuff that I rubbed onto their bare areas to help heal and stop further pecking, but it doesn't stay on well and didn't seem to do much for them. Plus, it was kind of expensive, considering I have 9 chickens to put it on and would have to apply it probably daily to have it deter pecking. What would you do? Should I separate the 4 that have all their feathers and hope the rest get better...assuming the 4, or at least one of them, are the mean one(s)? I don't want to cull all 4 when I don't know if they are all doing it- although they all look nice and pretty with no bare spots at all, which is suspicious.
All I know is I feel so bad for the others! I thought the roo would be the top of the pecking order, but apparently not- since he's getting the worse of it. It started a little after they reached maturity and the roo started doing his business, so at first I thought the bare backs were from him, and it may have started that way, but now there is pecking happening and it's just getting worse.
Thank you so much for reading this and for any input, advice, help you can give me! I appreciate it.

Issue #1- I have at least one egg eater. My question is what to do about it or how to find out who it is so I can take care of it. I still get a nice number of eggs, so I'm not too worried about it, but I would like to stop it before it gets worse or others catch on and start doing it as well.
Issue #2- and my biggest concern- All but 4 of my hens are being pecked/feathers picked. Even my roo. I only have 4 that look nice, all of the others have bare backs and some (like the roo) have bare spots near the neck as well. Sometimes they have scabbed spots and sometimes it's just red and bare. I tried what they feed store gave me- Rooster Booster- that was a purple cream-type stuff that I rubbed onto their bare areas to help heal and stop further pecking, but it doesn't stay on well and didn't seem to do much for them. Plus, it was kind of expensive, considering I have 9 chickens to put it on and would have to apply it probably daily to have it deter pecking. What would you do? Should I separate the 4 that have all their feathers and hope the rest get better...assuming the 4, or at least one of them, are the mean one(s)? I don't want to cull all 4 when I don't know if they are all doing it- although they all look nice and pretty with no bare spots at all, which is suspicious.

Thank you so much for reading this and for any input, advice, help you can give me! I appreciate it.