New home owner - previous owners left their chickens!

If it were me I would definitely remove the existing roosts and nest boxes and whatever that is that the nest boxes are on. A ladder style roost like Neal posted above would work well in your space and solve Rosie's issue with getting up and down. You might be able to reuse the nest boxes but they are simple to construct. Lots of people use 5 gallon buckets, milk crates and such also.

You will definitely need more permanent ventilation. @aart's post on page 2 has a link to a great article on ventilation.

That room is tight...it's what, maybe 6' wide?
Did you measure the 3", 6", and 3" with a tape measure?
The doors make it even tighter...you'll probably have to rebuild or replace that 'front' door, you might even think about making it open outward.
Not a good place for a ladder roost.... or even a ramp.
You might think about just keeping that back door closed off, or remove it.....what a funny little area out there.

Before doing any 'remodeling' I'd look at a really wide view of the coop room and the outside areas adjacent to figure out what you need/want to do for the long term,
so you only have to do it once. They'll be fine for now, tho it's not ideal and probably a PITA for you do deal with.

Can you get some pics of the outside areas surrounding the coop room?

What is your climate? Putting your location in your profile can help folks give better answers/suggestions.
Some things to think about:
Will you have to deal with snow accumulation and loads?
How will you enter the chicken area to do daily chores?
Do you want/need a secure covered run, maybe even a solid roof to help with water issues?
Look at the drainage and run off in the surrounding areas.

Just some thoughts.

So much of what you are dealing with reminds me of what we had with our used coop. And your reaction is so much like mine was. We thought we had it made....a 6'x12' coop for $200 and the farmer that sold it to us was willing to load it on a trailer and bring it to us. I wish I had been on here asking questions when we started reworking it. My gut said a lot of things would need to be redone but I didn't realize how much or how time consuming until I already had my chicks. And they needed to be out of my house. aart gave you some good advice as far as thinking about how you want to use the coop. I was in such a frenzy to get chickens outside I had no clear idea of how I wanted to do things. Now I have a lot of tweaking I want to go back and do before winter hits.

Your jammed window reminds me of ours. Two really great windows facing each other on opposite sides of the coop. They weren't just jammed, they were nailed shut and one even had wood strips nailed on it to keep it from EVER opening. Two nice windows that should have slid back and forth and they were rendered unusable. I still can't figure that one out. In our hurry to get chickens in we redid those windows with HC inside and outside. Now we are working on a way to cover them for winter and still have ventilation which I now have a better understanding of. Just a pain that we could have addressed at the time if we had thought things through a bit more. But it was already getting hot and I needed air in the coop.

You already have them in a lot better situation than they were so just take your time and consider long term. You will be happier if you only have to things once.

That strange little "room" on the back....that one is going to keep us all scratching our heads! Isolation room maybe? Who knows.....