Pokemon-go – Can you set a name for your Gym

If a gym belongs to your own team it can hold one defending Pokemon for each gym level. So a level 1 gym can hold 1 defender, level 2 can hold 2 defenders, etc. The spots are basically on a first come first serve basis. To level up the gym you can do "training" battles with the Pokemon that are in the gym by clicking the boxing glove icon on the bottom right.

You can't swap a Pokémon with one of your own. You can however, under some conditions, swap your Pokémon with one at a friendly Gym. This is only possible when:

• The Gym is at the maximum level of 10
• All places in the Gym are occupied (10 Pokémon)
• You don't already have a Pokémon at the Gym
• The Pokémon you want to add has a higher CP value than the lowest

In that case, the Pokémon with the lowest CP value will be kicked out of the Gym and is returned to its owner. Source (pg. 6).