I have a coop inside my pre-existing barn. One SIGNIFICANT mistake that I made was that I made my ventilation from the coop go into the barn. This means that chicken dander is getting all over my tools and everything I have stored in the barn.
I am going to have to redo my coop to ventilate to the outside (have some open vent areas with hardware cloth going to the outside, yes, even in the winter). Then I need to seal up the vents that I put in the coop that connects to the barn.
You could just leave one of your windows open, but cover it with hardware cloth, as long as it is sheltered so that rain does not come in. Make sure to seal your coop off from the rest of your building.
To answer your questions-
1. roost - is the existing 2x4 ok?
You will want it to be sanded and round the corners to prevent chicken foot problems (bumblefoot)
2. nesting box(es) - are they actual boxes? is it OK to put a simple door behind them to extract the eggs? I envision a flat board with 2 boxes (4 hens) on the right side wall. How deep should the boxes be?
My hens lay in a 10 gallon Sterlite tub that sits on the floor with some hay in it. For 4 hens I would just make one box, but make it big enough that two of them can fit in at once. My understanding from those with more than one nest box for a small flock is that they all just use one anyhow.
3. water system - what works for water?
For my 3 hens, I still just use the chick waterer! It works fine, and lasts them about a day in the hottest part of the summer. In the winter it goes inside a heated dog water dish. I have my waterer on a narrow shelf (no litter on the shelf) which prevents them from kicking stuff into it.
4. feed system - do I need to build it in? should I have something I can fill from outside?
I still just use the chick feeder with a quart container. This lasts them 1-3 days depending how much they are foraging.
5. trap door - I think a small hole (how big?) on the left side under the window so they can go out into the coop. Does it need a latch that I open or should they be permitted access any time they want
If your run is predator proof, then it is great if they can go in or out anytime. If your run is not predator proof, then they should be locked in at night. My hens go through a 7x9 opening, but others seem to like bigger doors.
6. If I put a wall on the right side, do I need vents?
You should ventilate to the outside, as above.
7. Would it be wise to insulate the 3 walls?
Where do you live, how cold does it get?
8. Do I need a window at the back of the coop? Notice there's one to the right and one on the left wall.
If you live in a cold climate, I would install more south-facing windows to help with winter, especially if you can do it in a place that is shaded by a tree in the summer. I would decide where to put the windows based on (a) south orientation and/or summer shading (b) aesthetics of free or cheap windows with your current windows and structure.
There is a little mini-tour of my coop inside the barn on this video starting at around 1:35 -
