Are these correct? Are John and Lynn coming? I say . John or Lynn is/are coming. (I say "is" is correct here.) John and Lynn are coming. I say . He and Mike hate each other. I say . Good? Thanks.
All present and correct. I've rephrased each sentence to show how I'd work out whether it's plural or singular.
Subject/verb agreement is fine, but your punctuation isn't. "Are John and Lynn coming?" I say. "John and Lynn are coming," I say. "He and Mike hate each other," I say. And, "John or Lynn is coming," is an awkward sentence: Either John or Lynn is coming.