Do all rich bodyguards live at home?

That depends on the nature of private bodyguards. Full-time private bodyguards should always follow their employers. If they are private bodyguards in different periods, they will arrange their own accommodation at night without bodyguards.

Private bodyguards usually live with their employers and also live with their employers at night. If employers are at home, they will live with them. If employers stay in hotels, they will stay with them.