Molting & Half a shell? Photos included

Egg eating, whether by your hens or by a predator, would be my guess. In this case, if you found the egg outside, I'd think a crow. That's typical of what they do, take the whole egg, break it in half, eat the contents, and drop the shells under the tree they ate in. Even if you found the egg inside the coop it could be a crow or another predator, or one or both of your hens. Sometimes something else broke the egg, whether having another egg dropped on it, or whatever, and the hens eat it and parts of the shell.

If they're less than 5 months it'd be a little precocious for them to be laying but certainly not unheard of. But chances are they're older than you were told. Certainly the moulting shouldn't show on the rest of their bodies until they're fully grown, by which I mean not just having hit puberty, but being physiologically true adults. Also, the bumblefoot and scaly leg on your hen's feet are not overly common on young birds. I did have one cockerel I got in with terrible scaly leg but generally it's something only adults show because it takes a while to progress to being visible.

If just the hen's bellies and breasts are bare, it's likely the broody hormones at work. Even hens that never go broody still experience the hormones, and often have bare breasts. Moulting shows on the rest of the body too.

Your hen shows both bumblefoot and scaly leg and will need treating for it before they get worse. Chances are the other hen needs treating too.

I personally use pure Stockholm Tar for their scaly leg mites if I get a bird in that has them, I just cover the whole leg and foot once or twice and it soaks in, and the entire affected areas fall off leaving perfect, pristine scales underneath. S.Tar's a great healer and killer of infections, even Golden Staph varieties, ulcers, fox bites, etc. It's also a painkiller. It removes splinters or foreign bodies as though by magic, kills pain, and kills gangrene and other infections, and removes cysts without surgery, not even leaving a scar to show where they were. It seems to soak into damaged tissues and enables the body to heal better, and very rapidly, and the affected tissues are rejected painlessly leaving often scarless tissue behind. I get it at produce stores, and it's a lifesaver. Messy, but I'm not complaining, lol!

I haven't had an issue with bumblefoot which I put down to diet. I always feed fresh raw garlic a few times a week, one or two cloves per bird. The sulfur levels over a few months build up to levels that act as a shield for the bird, buffering their immune systems, aiding rapid healing and regrowth, and sulfur is toxic to very tiny organisms including all internal and external parasites, so once they're protected by garlic scaly leg and bumblefoot cease to be issues.

This has been my experience, but many disagree. They used cooked or dried or pickled garlic because it's quicker than doing up a freshly crushed or cut few cloves a day and they didn't get the same results. If you're also unable or unwilling to put in the extra time a day (no condemnation here lol) then someone could advise you on some quicker acting chemicals to use. But you need to keep reusing them every time your bird catches the mites, because it's not permanent protection, and you'd also then have to treat the whole coop.

Your bird doesn't have severe scaly leg or bumblefoot but treating through diet I have found to be the easiest way as you don't have to treat the coop. A general good precaution against diseases and parasites is to feed raw garlic and use hydrated agricultural lime on the grounds they live on, as it destroys parasite oocysts and pathogens in the soil where they can build up or lurk for decades sometimes.

Best wishes.