HI, I dealt with a neighbor issue not unlike yours. Township, HOA, city ordinances...and court. If you'd like some advice from someone who won and actually got paid out after for all the hassle, here you go:
1. know your laws. All of them. Even the tiny ones. There's always a loophole for you. If it's not on the books, it cannot be enforced.
2. (And for some people, this is fighting dirty, but do what you must) Get your birds certified as ESA's...Emotional Support Animals. Look here:
http://www.servicedogtags.com/
3. Have your doctor confirm that you or someone in your home needs them for peace of mind and well-being of mental health. (Not a lie.)
But that way if it goes sour fast, the city cannot touch them...no one can. Legally you and they are protected.
Because what happened after the neighbor realized we'd complied with everything and our out-buildings were within code is that he then attempted to have the chickens taken from the property under the nuisance ordinance.
With them protected, HUD advocated for us and we won...and they're never allowed to bother us about our hens ever again.
Regarding the structure: Your neighbor should know that if you take the roof off and cover it with wire, it'll allow rodents in and around the property. (Truth. Sadly.)
Having a roof on there is MUCH preferable to having rats pass in and out all night long...and go over to his house as well, but hey. People as asshats, so if you have to remove the roof:
Lay down the wire and make it SMALL. I'm telling ya...stuff gets in. Norwegian rats can compact themselves very tight and they have a litter every 10-16 days.
Put clear plastic roofing material UNDER that to keep the coop dry. No dry coop and you risk disease and all kind of issues.
Over all of THAT, put a heavy duty tarp to block the sun and snow. If they hated the roof, they're gonna crap over a tarp, but this is their call.
Then if you have to get creative with something that isn't reliable in wind or snow, sure. Do the umbrella thing. But I'd layer up the wire and clear plastic roofing, personally. Keeping them safe and dry is the most important thing.
Always be very, very nice and professional to everyone you're dealing with. You know who told me OUR loophole? The city code enforcer.
Make sure you keep your coop super clean and smelling good, too. Don't give anyone any reason to complain.
And lastly, hope your neighbor moves or dies.
People who take out their own problems, hurts, and anger on others and innocent animals need to be removed from this world to make it a happier place. Ideally. Won't happen, but hey.
Don't forget to breathe. "This, too, will pass." Eventually, it'll end. I promise. Just know that you're in a bit of a hot war right now and you'll have to fight it.
At some point, the city is going to get very tired of this and stop entertaining the complaints, as it's a waste of their time and resources. Make sure they're on your side.