Why did he say "Elokei ha'Shamayim"?
Ibn Ezra: He did not say Elokei ha'Aretz, for Hashem gave the land to people - "Kol Mamlachos ha'Aretz Nasan Li Hashem."
Malbim: He shows that he believes that there is G-d in Shamayim who rules over everything. However, he thought that G-d appoints someone in place of Himself to conduct the land; G-d gave kingship over the land to him.
How did Hashem command Koresh?
Rashi: Yeshayah already said (44:28) "ha'Omer l'Choresh Ro'i... li'Yrushalayim Tibaneh v'Heichal Tivased."
Why must the Bayis be in Yerushalayim?
Malbim: That is where He desired to dwell, amidst the nation that serves Him.
Why does it say "bi'Yrushalayim Asher bi'Yhudah"?
Rashi: Yerushalayim is in Medinas Yehudah.
