will they be safe with the rooster and other hens?
You do not get guarantees with living animals about their behaviors. You just don't. Anything can happen. There are stories on this forum where a rooster did attack chicks. It is rare, but it can happen. Like Dobie, I have never had a rooster even threaten chicks, whether with a broody hen or hatched with an incubator and raised without a broody. Most of my roosters seem to ignore the chicks but I have had different ones help a broody with her chicks or help chicks after they were weaned.
There are other stories where hens attacked chicks. That is not as rare but my broody hens have protected their chicks. That doesn't mean that every broody hen will always protect her chicks from other hens but mine always have.
One thing that concerns me is when you say "It's not a large coop. It comfortably fits 4 English Orpingtons." To me, that sounds tiny, maybe 4' x 4'. If it is anywhere nearly that small I applaud you in finding a way to section an area off for the broody and the chicks. Is that coop at ground level or is it elevated? How big is the run, I think that is important.
I have an 8' x 12' ground level coop and over 2,000 square feet available outside. I think having a lot of room makes a big difference in how I see mine interact. I hesitate to offer an opinion until I know a lot more about what you are working with. With living animals all I can give you is an opinion, certainly not a guarantee.
Even if room is tight, following Dobie's suggestions will probably work. Many people do it that way, but the tighter the space the more likely you are to have issues.
I totally agree with leaving her where she is with her chicks. I like Dobie using 48 hours instead of 2 days. Several people think 2 days means 25 hours. But then you have a decision to make. Tear down those barriers or move her and her chicks.
I never build those barriers. I let my broody hens hatch in the nest and manage their chicks. But your coop is smaller, I think those barriers could be a good idea.
It will be a while before your walk-in coop is ready if I understand it correctly, so it is too late to count on it for now. What does that smaller coop and run look like? How is it set up relative to your main coop and run? One option would be to set it up so both runs share a common fence. Lock the hen and her chicks in the smaller one for three days then provide a gate or hole so all of your chickens can go anywhere. The broody hen should have enough room to work with to keep her babies safe. She should take the chicks to sleep in that small coop at night. She should be able to integrate her chicks with the rest of the flock so they will not try to harm or kill the chicks, either now or when she eventually weans them.
It will be simpler if you can just tear that barrier down in a few days. If that works it will be great. I very much agree you need to introduce those chicks with the main flock as early as you reasonably can. If you need to use that second coop while you finish that walk-in, I'd set it up so they can integrate at a young age.
