Continuity mistake: When Furley enters The Apartment at the start of the party, the outside weather is crystal clear. Less than 2 minutes later, Furley tells "Austin" the fog is very heavy and the airport is closed down. Then upon opening the front door, tons of thick fog is all over outside. While fog can roll in very quickly, how could Furley even bring up the fog in the first place when it was not there at all when he walked in?
Character mistake: When Jack tells Mr. Furley his brother is in town (before mentioning it was a "twin" brother who Jack made up), Mr. Furley gave a puzzled look and said "Your brother?" which appeared as if he didn't know Jack had a brother. However, in the previous season, Mr. Furley briefly met Jack's brother Lee, so he already should have known Jack had a brother (even if were to turn out to be a different brother).
Answer: He's not a Catholic priest. He's a pastor, with the title Reverend, at a community church who chooses to wear a clerical collar. Some people still address pastors as "Father" though. I'm not sure of his religious denomination, but most pastors can marry.
Bishop73