The speech Artie gives to guide Jim to the source of the sounds is the opening sentence of Thomas Jefferson's Declaration of Independence.
The saloon set that serves as a courtroom is the same set used as the Wayfarer's Inn on Cimarron Strip (1967).
While some sources categorize burgoo as a Mulligan stew type dish to truly appreciate Artemus' reverend's reaction to it look to the article There's Never A Recipe For Burgoo.
The "gunman" runs up the stairs to "Doc" Galen Adams office from "Gunsmoke" and the judge is Paul Fix from "The Rifleman" and dozens if not hundreds of other Westerns.
As West and Gordon chase the shooter down the street the facade of the Barkley mansion (The Big Valley) can be seen ahead and on the left. This facade was used many times in Wild Wild West.