How were the tribe, family and sinner identified?
Radak (13) #1: Yehoshua passed them in front of the Aron. The one who was caught, the Aron trapped him, and he could not move from there.
Radak (13) #2: Yehoshua stood all the Shevatim in front of the Urim v'Tumim. The names of the Shevatim were written on the stones. The Shevet that is caught, its stone will fade. Yehudah's stone faded. A lottery revealed the family and person.
What do Mishpechos, Batim and Gevarim refer to?
Rashi: (Bayis is a part of a family.) There are many Batei Avos in a family. Gevarim are the heads of the households.
Radak: These refer to the heads of the families, Batei Mishpachah and people of the house.