Team Mode is unlocked at lvl80. In this mode you can select a team to play together on game modes like Team Boss.
Bonding is a way to power up that is unlockable very early in the game. You can bond multiple Pokémon to your main Pokémon, in order to give your main Pokémon some extra stats.
Cross Server Gyms are unlocked at lvl70 together with Z-Rings and Treasure Raid. In this game mode your goal is to defend a position on each Gym in order to earn Factory Stamps which can be converted into Z-Stamps.
Z-Rings are unlocked at lvl70 together with Cross Server Gyms and Treasure Raid.
Treasure Raid is unlocked at lvl70 together with Cross Server Gyms and Z-Rings. What are you supposed to do here? Nobody knows.
Each of those Genes provide stats, and they can be upgraded with items that you obtain from the Sky Pillar.