Computer Go algorithms applied to other games?
Has the Monte Carlo method already been applied to other games? (Are there concrete implementations available?

Yes. This Grad paper might be of interest to you. It covers Backgammon, Bridge, Go, Scrabble, and Clobber. 


Backgammon, implementation TD-gammon.
Bridge, implementation Bridge Barron. Probably outclassed these days by other computer implementations.
Scrabble, implementation MAVEN.


For which kind of games, among those that have clearly stated rules and won't rely on complex communication between players, this method can be expected to fail?

Actually, Monte Carlo methods don't do very well with end games in general. You will find that many computer implementations that use Monte Carlo methods employ different algorithms when playing the endgame.


MAVEN uses B* search algorithms during the endgame (no more tile to draw).
MoGo uses 3x3 patterns to help identify "interesting" moves, and Upper Confidence applied to Trees (UCT) to help prune less optimal random moves from consideration and speed up searching.


While Monte Carlo methods are a useful tool in playing board games, they aren't a perfect solution. Computer Go programs using MCTS, are still unable to beat the best players on 19x19 boards without significant handicaps. Computer Bridge programs are no match for expert players. Monte Carlo methods will only approach perfect game with infinite time/space.