That would work as long as there's enough overhang to protect the nests under it. Keep in mind you'd then need to scrape poop off the nest tops.
Personally I'd rather move the roosts and avoid any chance of poop splattering into the nests.
Yes definitely needs more ventilation, the one window I can see appears to be glass/plexi and not open for ventilation.
The roosts are badly placed over the top of the nests like that - nest boxes under roosts need a good amount of overhead protection..
The overall design isn't bad, though I'd switch up the roosts and maybe do 2 parallel roosts just above nest boxes in height. You'd only need 2 nests though for 4 birds.
Other change is I'd add a LOT more ventilation on the inside coop wall facing into the run, like a giant "window" or two that...