To qualify for the batting title, a major league player needs 3.1 plate appearances per game played. In a standard 162 game season, that comes out to 502 plate appearances.
For a pitcher to qualify to the ERA title he needs one inning per game played.
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Rule 10.23a of the Official Baseball Rules: MINIMUM STANDARDS FOR INDIVIDUAL CHAMPIONSHIPS 10.23 " To assure uniformity in establishing the batting, pitching and fielding championships of professional leagues, such champions shall meet the following minimum performance standards: (a) The individual batting champion or slugging champion shall be the player with the highest batting average or slugging percentage, provided he is credited with as many or more total appearances at the plate in League Championship games as the number of games scheduled for each club in his league that season, multiplied by 3.1 in the case of a major league player. EXCEPTION: However, if there is any player with fewer than the required number of plate appearances whose average would be the highest, if he were charged with the required number of plate appearances or official at bats, then that player shall be awarded the batting championship or slugging championship. EXAMPLE: If a major league schedules 162 games for each club, 502 plate appearances qualify (162 times 3.1 equals 502). If a National Association league schedules 140 games for each club, 378 plate appearances qualify (140 times 2.7 equals 378). Total appearances at the plate shall include official times at bat, plus bases on balls, times hit by pitcher, sacrifice hits, sacrifice flies and times awarded first base because of interference or obstruction. " To answer your question, take the number of games the team has played at the end of June and multiply by 3.1. If the player has that many total at bats, (team total games * 3.1), then he eligible to be considered in the batting title race.
There is no official minimum for the MVP award. The minimum to qualify for the batting average title is 3.1 plate appearances per game, or 502 for the season. Regarding the MVP vote, more plate appearances give a player more opportunities to establish value, but there's no specific minimum for eligibility.
The eligibility rules for rate statistics don't stipulate number of at bats, but rather plate appearances - 3.1 per game played by the team. For a standard 162 game schedule, this comes to 502.2 plate appearances, meaning the batter must have 503 plate appearances to be eligible.
However, if he went 1-1, then walked the rest of the 502 times he batted, he would still win the batting title, because he had enough plate appearances!
According to MLB, a rookie is "A player who hasn't accumulated 130 at-bats or 50 innings pitched in previous seasons and hasn't spent 45 or more days on 25-man active rosters, not including times when the active list is expanded to 40."
I do not believe there are any hard numbers as far as how many at bats or how many innings pitched a player must have to qualify for selection of ROY. The only qualification is what is listed above for a player to be considered a rookie.