That looks workable. I would change a few things.
You'll want to leave some space behind the rack for running cable and airflow. If you can give yourself a little more space, plan for a 1m rack. Those are very common. You can easily find used ones. A typical nas chase will be a tight for a 800m deep rack. I have this one:
Super Huge Storage 24 Bays 4U Hotswap Rack NVR NAS Server Chassis in my 900mm rack, and it feel just right given all the cabling you have to do in the back. In hindsight, I wish I had gotten something like this:
24 Bay 4U Server Chassis, Hot Swap HDD Rack, which would have been better at keeping my hdds cool. You can probably get rid of them easily too as they are pretty standard. I have 90cm rack which was probably twice as expensive as a 1m rack, but my office is small and I need every inch (or cm haha). I'd keep a little gap at the bottom of the door so cool air can come in. Actually if you can get a vent to pump in cool air in the closet some how, that would be awesome. You just want to make sure it's far from the exhaust, so it doesn't get sucked out.
If you can give yourself some room on the sides of the rack that would be great so you can get behind the rack. You'll have to go back there for unhooking servers and doing cable management. You'll also need space for yourself and any servers you want to take out. I think the deepest consumer server case I've seen about 67cm. So you'll need space for that and yourself in there. If you plan to take the server rack out for cleaning under and behind, you'll want doors big enough to move it out as well. Keeping the area around your sever will go a long way towards keeping your equipment healthy.
One thing I'll say is that you'll never regret having too much space in your server room, but you will regret having too little. Oh yeah, see if you can have them install a fan in the exhaust vent. I'm wasn't sure if that was implied in what you said with respect to the vent. They aren't expensive, but they can be critical for removing heat. If you buy decent fans, I wouldn't worry too much about the noise in your setup.