What is the meaning of "v'Hacharamtem"?
Radak, Targum Yonasan: It is an expression of wiping out, like "v'Es Kol ha'Neshamah Hecherim" (Yehoshua 10:40).
Radak citing his father, Malbim: It is an expression of Cherem and a curse. First make a Cherem, so no one may benefit from them, and afterwards kill.
What are the reasons why we must destroy Amalek?
Malbim (2): All [other] wars are for one of five reasons. (a) To conquer land, or lest one's land be conquered. They attacked while you were on the road! (b) They fear lest others nearby pass through their land, like Sichon and Og. You were far from Amalek! (c) There is a quarrel or hatred between the nations. They had no previous dealings with you! 1 (d) To show their strength. Amalek did not. They attacked the weak of Yisrael! (e) They fight a nation that denies their creed or deity. Amalek did not fear Elokim! 2 Rather, they attacked to deny Hashem! After the wonders that Hashem did for Yisrael, they wanted to show that He cannot save them. The war was primarily against Hashem, and eternal hatred against the Rock from which Yisrael were quarried - "Milchamah la'Shem ba'Amalek mi''Dor Dor" (Shemos 17:16).
Perhaps they bore hatred for Yisrael, for their matriarch (Timna) wanted to convert, and Avraham, Yitzchak and Yakov all rejected her (Sanhedrin 99b)! Perhaps Malbim means that normally, the hatred was due to a quarrel. (PF)
In Devarim 25:17, Malbim says 'they fight to find favor in the eyes of their god by destroying those who do not believe in it.' He understands that Amalek did not fear any Elokim. Refer to Devarim 25:18:5:3*. (PF)
Why did he say "do not have mercy"?
Radak: In the end, he would have mercy - "va'Yachmol Sha'ul veha'Am" (9). Therefore he was warned, so he would not have an excuse.
Why does it mention man, woman, child and baby?
Malbim: This is in decreasing order of strength. (You must kill even those for whom there is no concern lest they fight us.)
Why were we commanded to destroy even their animals?
Rashi: Amalek were witches, and could change themselves into the form of animals. 1
Radak: If an animal remained, it is from the spoils of Amalek, and their remembrance was not wiped out. 2 We are commanded "wipe out the remembrance of Amalek" (Devarim 25:19)!
Rashi implies that if not for this, we could have kept the animals, i.e. we could take [other] spoils from Amalek. Refer to 15:3:5:2 and the note there, refer to 15:12:1:2 and the note there. (PF)
Presumably, the same applies to all spoils; it is Chidush that the animals must be killed, even though "v'Rachmaav Al Kol Ma'asav" (Tehilim 145:9), and some say that the Torah forbids Tzar Ba'alei Chayim (Shabbos 128b). Also Abarvenel and Ran (Sanhedrin 43b) hold that all spoils were forbidden. Refer to 15:3:5:1 and the note there, refer to 15:12:1:2 and the note there.. (PF)