1. You combo a new gem and it activates. The better the power the less frequent the gems shows up.
2. There is no button for it. Keyboard configuration is already in the game.
3. You can swap the active companion on the overworld. You won't be able to do it in puzzles or combat
4. We don't have any plans to make some sort of jealousy system or locking off content because you picked one girl first. The only condition for not dating someone fully is how much exp they have before the end of the game.
5. Well, outside of learning more about the girl, you'll get new intimate animations and perhaps a new skin to use in the main game.
6. You can't swap companions mid puzzle. So there is no exp sharing.