Mark Zuckerberg likes his privacy, but he’s in an increasingly public battle with a would-be neighbor in Palo Alto, Calif., that threatens to expose details of his personal life and conduct. New court documents were filed on Wednesday in a continuing lawsuit against Mr. Zuckerberg, the Facebook co-founder and chief executive, revealing a dispute that touches on various hot buttons of boom-time Silicon Valley: privacy, networking, precious real estate, and the pairing of youth and affluence. In one sworn document, a real estate broker for the man suing Mr. Zuckerberg testified that Mr. Zuckerberg’s own real estate broker referred to...