It would depend how handy you are as to whether you can build it yourself. You can buy the panels premade or construct your own. You could put a sloping roof on with the span not being very wide.
I consider myself as quite handy and have build quite a lot of things made of wood or zinc and copper sheets and so on in my life, thing is only, that I suffer from a lack of suitable tools since I came to UK
My old Makita hasn't the power for those brick walls, I already tried that just for a ridiculous washing line hook and could not succeed though the drill was the right one. And the same is for the wood sheets.
Wooden beams I can saw with a hand saw but a wooden sheet would take ages and my jigsaw seems not to be quite adaptable for British Jigsaw blades. So if you cannot borrow those tools from DIY shops as some in Germany offer, this could be my biggest draw-back. My first thought has been to get the framework at the open side, fix trusses around the brick wall's top for the roof construction so the slope roof could be turned outside the yard for the drainage (there's a wide enough green space between wall and pavement, so nobody will be disturbed but the yard gets less additional water from the roofs. Or it had to be a roof with a gutter and could end in a rain barrel.
The coop definitively must be back side to the tree as on hot summer days it gives the chooks additional shade as for from late afternoon in summer there will be sun in this corner almost till dawn plus it gives more wind protection... However I found car covers quite usable for the problem of draught with the rabbits. It lets air through but is at least 95% waterproof, so even on a sunny day it will not heat up much more than the outside and the rabbits have a bigger shelter and their kennels stay dry, too. Looks a bit spacy my "Mars Station" that I told my husband is a Gas Chamber undercover...
But climate is perfectly fine underneath. Just to give you an impression what I mean, there is a picture...
However, those car covers might be another way for summer heat protection and especially for draught protection.
So, all I really would need is: A way to borrow those tools without having to buy them yet, and someone handy to help either to keep my arm extensions happy and entertained or giving me a hand with the building (or both). Ideas I would have enough...