Has any girl trainer won the league in Pokémon or do girls and boys go in separate leagues?
In the core series games, the player character wins the league and can be chosen by the player as male or female (except in RGBYGS, where the character is male). Before the player wins the league, it has a champion who is in some cases female (Cynthia, Iris, Diantha).
In the anime both sexes compete in a common league, which so far is only known to have been won by males (an unknown male in Kanto, plus several named males in the later leagues). However, contests often have female winners, while showcases only have female entrants.