I agree, do not house goats in the same building, pretty much for Aart's reasons. Your challenge may be to deny the goats access to the chicken's coop, they can get through some pretty small openings.
I put both the nesting boxes and roosts up high. Probably about 3.5 feet.
That is not even close to high. One issue may be that chickens often like to sleep in the highest place available. You want your roosts noticeably higher than any other place you don't want them to sleep, like your nests. If the roosts are right next to the nests 6" higher can be noticeable. If across your 6' x 8' shed 12" should work. You don't want them sleeping in the nests when they start to lay, you don't want poopy eggs.
Some people really freak out at the thought of chickens sleeping above the nests. That does not bother me in the least as long as the nests are covered so the poop cannot go in the nests. I put a juvenile roost over my nests so the juveniles have a safe place to go when they are ready to roost and the adults won't let them onto the main roosts. I use the top of the nests as a droppings board when juveniles are up there. They are obviously higher than the nests. It works well for me.
I have EIGHT 10.5 week-old easter eggers. So far they don't seem to have any interest in exploring the roosts or nesting boxes, but they are also still adjusting.
You don't have adults which can change their behavior. And yours are brooder-raised, which is different from broody-raised. When no adults are around my brooder-raised chicks tend to start sleeping on the roosts at around 10 to 12 weeks old. I have had a group start as early as 5 weeks, I've had some go past 12 weeks but 10 to 12 is a good average. I think what my coop and roosts looks like plays into that. Each of us are unique and certainly do not get the same results. You just put them in so give them time.
Many people on this forum seem to think something is wrong if they aren't immediately roosting. They will be out there setting them on the roosts to train them to sleep on the roosts. I take a different approach. When they are ready they will roost. Until then they sleep where they want to sleep. As long as it is not in my nests and is somewhere predator safe I don't care where they sleep. They are safe and it won't hurt them.
I do have a climbing ladder, but am worried now that everything too high and could cause injury. Appreciate any thoughts on what easter eggers might prefer and what's the maximum height for safety.
Easter Eggers are not a breed, there are no standards for them. They can be any size though usually they are fairly small compared to other dual purpose breeds. You have obviously read about how a chicken can hurt it's legs or joints if it hops down from a high roost. That's possible but let's look at the rest of the story. Overly large birds like Brahma's, Jersey Giants, meat chickens, or chickens fed so they are large for their breed (like birds raised for show) can possibly hurt their legs like that. It is a real threat to them. But that is not much of a risk if they are not overly large for their breed. My birds are hatchery stock dual purpose like Orpington, Sussex, and Australorp, probably larger than your EE will be, depending on how you feed them. I do not feed them a high protein diet so they don't grow that large. They do need a decent clear flight and landing area but mine have no issues flying up and down from a 5' high roost. They could fly higher but I don't have my coop set up that way.
I've had several hens and even a full sized rooster once launch from that 5' high roost, fly about 8' forward, hang a hard left, fly out of my coop human door, and land in the run. Chickens are not great flyers but as long as they are not oversized they do OK. This is why for your probably not oversized EE's I don't consider 3-1/2 feet to be high at all.
I like my roosts to be high enough off of the coop floor so the chickens cannot reach the chickens perching up there. When I integrate new chickens I often find them on the roosts out of the reach of the others. It is a safe place they can go.
But there are reasons I don't want the roosts too high. One is the higher they are the more clear area they need to safely fly down. In New York you will want ventilation in the winter to help keep your coop dry but you don't want a cold wind hitting your chickens on the roosts. A good way to achieve this is to have your ventilation up high so any winds through the ventilation are above them when they are on the roosts. Also, a good time to check them for mites or to other issues or to work on them for other reasons is at night when they are on the roosts. They are easy to catch in the dark. So I want the roosts a convenient height for that.
We are always interested in seeing photos so we can see what you are working with. From what you have written I don't see anything to be concerned about, except those goats.
