In our area coops must be placed 50' from buildings, per municipal code. Since I, too, live on a suburban lot that's just 35'x75', that's utterly impossible.
I tucked my coop up next to my house between mine & our neighbors (renters). The coop is, literally, on the property line. There's about 5 feet between our house and the property line, so we just enclosed that area as a run. It was already fenced but we never went back there. I intend to bribe the neighbors on that side with eggs once the girls do start laying.
The girls free-range during the day only if I'm at home. That helps alleviate the problem of being so close. It also bring the problem of flies into my yard, front & center (yuck!).
As long as you're legally allowed to have chickens and are a good owner, there's no reason you shouldn't get them and put them where ever you want in the yard. Follow the local codes as much as possible and be the best neighbor you can be. If any complaints come and you're investigated, you'll be able to prove due diligence. Even though the neighbors "aren't into the chicken thing", it's your property.
BTW, re the dog thing - my neighbors on either side have dogs. I've got an agreement with each: keep your dog on your side and I'll keep my chickens on mine. If my pets wind up dead in your yard, that's my fault. But if your dog kills them in my yard, you'll owe me financial restitution and a better fence.