Why does the Kohen insert "Ki Hashem Elokeichem ... " here?
Rashi: To remind them that, whereas the enemy are going to war relying on the strength of flesh and blood, 1 you go to war relying on Hashem's strength (a good enough reason not to be afraid - Torah Temimah).
Ramban: To warn them not to rely on their own might and prowess, but on that of Hakadosh-Baruch-Hu. 2
Rashi: Like we find in Shmuel 1, Perek 17, when the P'lishtim went to fight Yisrael relying on the might of Golyas - and see what happened to him in the end!
Ramban: As the Pasuk writes in Tehilim, 146:3, 147:11 "Al Tivt'chu bi'Nedivim ... Rotzeh Hashem es Yere'av, es ha'Meyachalim le'Chasdo."
What is "Hashem Elokeichem "ha'Holech Imachem" referring to?
Rashi (citing Sotah, 42a): It is referring to the Machaneh ha'Aron. 1
Targum Yonasan: It is referring to the Shechinah - which went in front of Machaneh Yisrael.
Yerushalmi Sotah: Some say that it is Hashem's name, which was placed inside the Aron.
Yefei Mar'eh (in Yerushalmi Sotah 8:3): It refers to the Machaneh Leviyah.
Torah Temimah, which contained the broken Luchos and which Yisrael took with them when they went to war. See also Sifsei Chachamim.
What are the connotations of "Lehilachem lachem" and of' "Lehoshi'a eschem"?
Ramban: It means 'to destroy your enemies and to save you from their hands, so that not one of you will fall in battle'. 1
Ramban: Which explains why Yehoshua cried out to Hashem when, in the battle against Ay, thirty-six men of Yisrael fell - See Yehoshua, 7:7-9.
What is the 'Machaneh ha'Aron' to which the Pasuk is referring?
Sotah, 42a: It refers to the Aron 1 which contained the broken Luchos.
Maharsha (in Sotah, 42a): It refers to the Shechinah.
Yerushalmi Sotah: Some say that it is Hashem's name, which was placed inside the Aron.
Yefei Mar'eh (on Yerushalmi Sotah 8:3): It is the Machaneh Leviyah.
With reference to Machaneh Yisrael, who took the Aron with them to war (Torah Temimah).