Why did Hashem strike him with Tzara'as?
Rashi: It says in Divrei ha'Yamim II, 26:16-21 that he entered the Heichal in order to burn Ketores. Radak - Moshe hinted to this in Korach's rebellion - [Lema'an Asher Lo Yikrav
We do not find that Korach received Tzara'as! Perhaps this is 'included' in his death; a Metzora is merely like a Mes. (PF)
Where is "Beis ha'Chafshis"?
Rashi citing Yerushalmi Kidushin 1:7: He made a house in a cemetery. This is like we say "b'Mesim Chafshi" (Tehilim 88:6) - a Mes is Chafshi (free, i.e. exempt) from Mitzvos.
Radak: Kingship is Avodah, for he must bear the burden of the nation, hear them and judge their cases, whether or not he wants. When he became a Metzora and dwelled in a house alone, he was Chafshi (free) of this Avodah.
Radak citing Targum Yonasan: He dwelled outside Yerushalayim.
What is the meaning of "Shofet Es Am ha'Aretz"?
Radak: As long as his father was a Metzora, he [acted as king, and] judged the people.
Radak citing Chazal: Yosam was born the year that Uziyah became a Metzora. He judged the people only after he matured and had Da'as. Until then, the Chachamim and officers judged the people.