Few clues suggest a pro wrestler lives in Mick Foley’s Long Island home. A cartoon drawing of his leather mask-wearing alter-ego, “Mankind,” is pinned to the refrigerator. In the game room, a steel chair—the kind he used to swing at opponents’ heads—stands near a bed for Randall, the family’s Papillon.

Mr. Foley has dedicated more of his 4,100-square-foot contemporary to another of his colorful personas: Santa Claus.

On...