Looks like you have a covered run that is 10 feet long and 3 feet high, width unknown. The way that's drawn it looks like the 36" is height. It's probably actually 3' wide. But the way your current coop looks like it is set in the run 3' wide doesn't look right either. I'm confused.
I could not find an active Arizona thread in the "Where am I? Where are you!" section of this forum. It's often helpful to chat with neighbors when you are looking for something. Can you find a local facebook group or some other local social media group that looks like it might be a good place to ask about a building?
I don't know where you are in Arizona but your weather in winter probably isn't that cold as far as chickens are concerned. Your danger is going to be the summer heat. Your coop will need really good ventilation for summers. They will need shade. You probably have weather that they can be outside all day every day which makes coop size a little less important as long as they have access to the run when they are awake.
I am concerned about how I would connect the two (so that the birds can safely travel to the nesting area+run and back to the roosting area (colored in purple in the drawing).
Different ways. Some of it depends on how your current run is put together. One way might be to rip out that end of the current run and add to your run walls and top to tie in to that end of the building. You could make the run a little bigger (or a lot bigger) and have better access to everything.
Another way is to do something like this photo. This is a pop door between my main run and an area I have inside electric netting. Cut a hole in your fence and sandwich it between two boards screwed together. You can make it prettier than this, I just used scrap wood. By sandwiching the wire between two pieces of wood you greatly stiffen the fence and cover the sharp ends of the cut wire. If you screw the two pieces of wood together with screws in the holes in the wire and tighten them it is extremely secure. This gives you something to attach to if you want to attach to it or just give you a good opening.
You can put this at ground level like this and build a tunnel leading to your building out of wire only or wood and wire if you wish. I'd make the tunnel at least 18" high and probably that wide. You could use it as a pop door between two runs but I'd want access so I could get in there. The tunnel could be 6" long or 6'. There are a lot of different ways you could set it up.
If I were building or getting a new building I'd make it a complete coop. That means add nests as well as roosts. Sometimes hens lay right at the crack of dawn. Are they always going I have access to the nests out in the run? I think having the nests separated complicates things. That old coop is still valuable. It would make a great place to isolate an injured chicken if you ever need to, maybe help with future integration, or maybe handling a broody hen.