Hogwarts staff was created in 1997.
Remus Lupin and Horace Slughorn both used the Hogwarts Express to travel to Hogwarts. Otherwise, the only adults on the train were the driver and the trolley lady.
The headmaster; the teachers; some other staff such as the gamekeeper and the caretaker; the students.
A staff in Harry Potter could refer to the Hogwarts Staff (Teachers, employees, etc.) or it could refer to a walking stick such as Lucius Malfoy's staff/cane or Mad Eye Moody's staff.
It's very likely they did live in the castle as somebody has to look after the students and there weren't many other staff members. If they didn't live in Hogwarts they probably lived in Hogsmeade.
The Staff of Karnath was created in 1984.
Angel Staff was created in 2005.
Jack Staff was created in 2000.
The Alabaster Staff was created in 2003-07.
The Grand Staff
Every student left at Hogwarts as well as the staff, the majority of Ministry members, the Order of the Phoenix members, the creatures who live at Hogwarts and then just other witches and wizards who want to pay respect.
Special Staff F was created in 1941.
Britannia Staff Union was created in 1972.