It depends on genetics. Every chicken breed was developed by inbreeding. Practically every grand champion at a chicken show was developed by inbreeding. For thousands of years practically every small farmer’s backyard flock was continued mainly by inbreeding. When you inbreed you highlight what traits are there. If they are good traits, those get highlighted. If those are bad traits those are highlighted.
The trick is to select the offspring for breeding that have the good traits and don’t let the ones with bad traits breed. What are the good traits and the bad traits? Some are obvious like physical deformity. Some are up to you. If you want your hens to go broody, then hatch eggs laid by broody hens. If you want lots of eggs, hatch eggs from hens that lay lots of eggs, don’t hatch them from a hen that only lays occasionally. Things like that.
It’s pretty unlikely you are going to get deformed chicks. You are dealing with living animals with unknown genetics so no one can tell you for sure what will happen when they breed but you can normally go for four or five generations of inbreeding without any real problems as long as you don’t let obviously deformed chickens breed. You do have some responsibilities here. That’s the way the small farmers over the past few thousands of years have maintained genetic diversity in their flocks. Every four or five generation they bring in a new rooster.
Of course if you don’t incubate the eggs or let a broody hen incubate them you won’t get any babies.
A flock of juvenile chickens mating is generally not for the faint of heart. Hormones are running wild in the cockerels, the pullets generally don’t know what is going in since they mature later, and there is no adult supervision. It’s not just about mating either, there are a lot of dominance issues getting worked out. Normally they work through it and settle into a very calm peaceful flock when they finally mature but it can be rough getting to that point. Force is often used. Since it is rough it is possible one can get hurt but they normally don’t. With living animals no one can tell you what will happen for sure.
I forgot to mention, even if the chicks came from the same hatchery they are probably not that closely related anyway. Most hatcheries use a system to ensure long-term genetic diversity.
