No the Stanley Cup has never been decided by a shoot-out. It is actually impossible because in the Stanley Cup Playoffs overtime has a different format from the regular season. In the playoffs when a game is tied at the end of regulation (3x20 min periods and score tied) the teams enter sudden death overtime (which means the first team to score wins), but the overtime is very similar to regulation play. The teams skate 5-on-5 like a normal period and the periods are 20 mins long, also like a normal period. The game continues until one team scores. This means that theoretically if no one scored the game would go on forever. The most recent memory that comes to mind is the 1996 Stanley Cup Final where Uwe Krupp of the Colorado Avalanche scored the cup-clinching overtime goal in the third period of overtime (the sixth period of the game).
Montreal canadiens
Two
5-5 is played with 20 minute period of overtime. this is sudden-death, however.
The Montreal Canadiens and the Calgary Flames.
Marian Hossa (now of the Chicago Blackhawks), played in the 2008 Stanley Cup Finals for Pittsburgh (and lost), played in the 2009 Stanley Cup Finals for Detroit (and lost), and played in the 2010 Stanley Cup Finals for Chicago (and won).
According to Wikipedia: "When a game goes to overtime, each team is given one possession from its opponent's twenty-five yard line. The team leading after both possessions is declared the winner. If the teams remain tied, overtime periods continue, switching the order of possessions for each overtime, until one team leads the other at the end of the overtime."
The Dallas Stars defeated the Buffalo Sabres in 6 games.
Montreal Canadiens - 4 vs Chicago Blackhawks - 3
27 different teams
Detroit Red Wings and Pittsburgh Penguins. Detroit won.
Boston Bruins over New York Rangers 4 games to 2.