Yes, they have done studies on this. There have been all kinds of studies done to try to determine what teams are successful in the postseason. The factor that correlates strongest to postseason success is regular season success, but even that is not a great predictor.
As far as the momentum thing goes, Baseball Prospectus did a study analyzing the correlation between a team's performance in the final week, the final 2 weeks, the final 3 weeks, and the final month of the regular season and a team's postseason performance. None of the correlations even reached .05. In other words, the relationship is random.
There is also no carryover of momentum into the next game after a team has a big comeback win late in a game.