If you win the tournament on hard, you can get all the pokemon (including all pokegods).

Pokémon Stadium Cup requirements:
Pika Cup
Pokémon can only be between level 15-20.

Petit Cup
Level ranges between 25-30. There is a height requirement though. Height 6'8" limit and weight 44 pounds.

Poké Cup
Only can use 3 Pokémon, levels 50-55. The sum of the Pokémon's levels cannot exceed 155.

Prime Cup
Everything is legal. You can have any level, height, weight, or number of Pokémon. However, your opponents' levels are always 100.