You are really lucky in that your landlord built you such a nice coop. I like your modifications you posted. Lots of good suggestions from others, too.
For nest boxes, I use some Dollar Tree dish pans...
They are cheap, plastic, and easy to clean out if/when you need to. In your case, it looks like it would fit into the next box compartments. The walls of the dish pan will keep the eggs inside in one place instead of rolling all over the place.
So, the next things we will be working on are
1) walls inside...
I don't think you need inside walls. I agree with others. It's just asking for more trouble.
4 Flooring... Would it make sense to put in a pvc/vinyl flooring (removable for ease of cleaning) or would something (like plywood?) that could be painted be better?
I put down a vinyl sheet on my coop floor and it has held up for about 4 years before it started to crack and rip. I guess it has done its job and should not complain.
If I had to do it over again, I think I would just lay down some sacrificial wood on top of the flooring and replace it as needed. But that is because I now have lots of free pallet wood to work with. It would not cost me anything to line the coop floor with pallet planks and they would be easy to replace.
One thing I really like in my coop is that I use deep dry bedding. I have maybe 4 inches or more of litter in the coop at the start of every winter. I don't use poop boards under the roost. I just let the poo drop below and once or twice a month I toss on more fresh litter into the coop and especially under the roosts. The poo will automagically disappear into the deep bedding, dry up, and not cause a problem. I only clean out my coop every six months. And that seems to be enough. Of course, your nose will tell you if you need more fresh litter or if it's time to do a complete clean out. Depends on how many chickens you have, of course.
Another thing about using dry deep bedding, in my coop, is that the floor never gets "wet" from the chicken poo. All that moisture is absorbed into the deep bedding before it reaches the flooring. That is why I don't think I really needed to put down a vinyl flooring at all.
As someone else mentioned, I am not into oils or other preservatives in my coop, either. I prefer to keep everything as chemical free as possible. I would rather replace a sacrificial board once every 4 years than adding chemicals to my coop.
I like how you made the front door bigger. I would just rake out all the coop deep bedding through the front opening into the chicken run and let it compost in place. If you use enough deep bedding, it should not smell. If it does smell a bit, cover it with fresh leaves, grass clippings, etc... Then just let the chickens work the run litter and break it down into compost for you.
I converted my entire chicken run into a chicken run composting system after my chickens destroyed all the grass in the run and left nothing but bare dirt. The chicken run compost works better for me anyway because I have lots of raised bed gardens to feed with compost.
Love to see your upgrades to your coop. Thanks for sharing.