There is no minimum badge requirement to come and visit Mysterious Treehomes. Simply send Lusamine a PM or get permission from me in the chatbox and you're well on your way to a secret Pokemon adventure! Once inside, you're free to explore any area of your choosing.
In regards to the house entryway pokemon, they are a protected species and therefore cannot be caught. They will grow with you, as a trainer, becoming stronger and remembering their past interactions to develop a relationship based on how they were treated.