Is there any halachic basis a beis din could stop a developer from building on his own property because neighbors don’t want the development?
Yes. Perek Lo Yachpur and the halachos of Nizkei Schanim discuss when you may not build on your property due to the adverse effect it will have on the neighbors.
Having never been called to Beis Din (Beli Ayin Hara), how does that work? If a Bais Din in Lakewood calls me in, can I respond with a suggestion to go to a Bais Din in Chicago instead?
You can not ignore a hazmona but you don’t have to litigate in the BD that issued it. You can suggest a different BD. If the plaintiff doesn’t agree to your BD then each of you can pick a dayan and those two dayonim pick a third dayan to form a zabla BD.