You're right to point out that these awards come from different organizations and that each may have different rules, but if you're looking for a general rule of thumb: to win an award for either the National or American league, a player should play the majority of the games for that season in that league.
Even meeting the criteria of having played a majority of games won't make it easy though. "No player switching leagues has ever won an MVP award and only one, Rick Sutcliffe of the 1984 Cubs, has won a Cy Young. Sutcliffe went 16-1 in 20 starts after the Cubs acquired him from Cleveland".
For the MVP award, the number of games played is actually the number 2 criteria given to voters (I'm assuming that means number of games played in the given league).
There are specific qualifications to be eligible for the Golden Glove awards, for example
All pitchers must have pitched in at least 137 innings by his team’s
137th game
All infielders and outfielders must have played in the field for at least 690 total innings through his team’s 137th game: this equates to playing in the field for approximately 7.5 innings per game in approximately 67% of his team’s games by his team’s 137th game; this ensures that only full-time players are considered);
It doesn't say it explicitly, but I assume that by his team, they kind of mean for it to be a single team in a specific league to qualify for that league.
As to whether or not voters can consider performance in the other league that happened before a trade, I found a couple articles about Manny Ramirez and CC Sabathia, both traded in 2008, that suggest that only games in the awarding league matter.
