Can minor league players be called up during mlb playoff games?

Answer 1

Not unless there's an injury that forces a player to be completely removed from the roster.

At the start of the first game of each playoff round, the manager must submit a list of 40 players he wants to be eligible for this round. This list can (in theory) include players with no major league experience*. If a player is injured, he can be removed from the roster and replaced by someone else -- again, this can (in theory) include someone with no major league experience. However, the player removed due to injury can not play in any further playoff games that season.

If the entire team slumps (like the 1966 Dodgers), that's just too bad.

* Note that any player put on the post-season roster MUST have been part of that team's organization on or before August 31 of that year -- no ringers allowed!