While "Plane Crazy" was the first Mickey Mouse cartoon produced, it was not officially released on May 15, 1928. According to Neal Gabler's Walt Disney biography, "Plane Crazy" was finished sometime before May 15th and Walt immediately started shopping it around to local Southern California exhibitors in hopes of finding a national distributor. Gabler cites individual preview showings in Glendale and Hollywood prior to May 21st. Despite favorable audience response (one exhibitor recalled the cartoon getting more applause than the feature), no distributor showed interest. By then "The Galloping Gaucho" was already in mid-production, a silent cartoon like "Plane Crazy."
Shortly thereafter, Walt made the decision to shelve "Plane Crazy," complete the animation on "Gaucho" and immediately begin production on a new Mickey Mouse sound cartoon. Accounts differ on how this actually came about, but it was clearly influenced by the recent success of The Jazz Singer with Al Jolson.
"Steamboat Willie" debuted at the Colony Theater in New York City on November 18, 1928, initially scheduled for a two-week trial run. The audience and critical response was overwhelming and the rest, as they say, is history. "Plane Crazy" and "Gaucho" were released with sound in December.
Film historians and the Walt Disney Company itself recognize November 18th as the "official" birthday of Mickey and Minnie, so I think we're safe to continue doing so without needing to pull hairs. It's amusing to note that while Disney fans the world over recall the significance of November 18th, Walt himself had difficulty remembering it. In several different interviews during his life he cited July 19th, September 19th and September 28th as the dates of Mickey's "Steamboat Willie" debut.
Edit: Needed to add that Pluto didn't make his screen debut until the 1930 Mickey cartoon "The Chain Gang."