How to unlock all the stars on the Trainer Card

Bulbapedia's pretty good on this, but I'll try to answer some of your questions here. When you switch blocks they'll stay like that until you switch them again. They 'level' by keeping that block out for 10 days, and some Pokemon only appear when you put certain objects/blocks of certain levels into the Safari Zone.

Joining a group is a feature introduced in Generation IV. Players in the same group encounter the same swarming Pokémon, weather conditions, changing Pokémon in the Great Marsh, Feebas location, and other things each day. Group members can compare records on the third floor of Jubilife TV.

Source: Bulbapedia