What is the meaning of "Ki Yad al Keis Kah ... "?
Rashi, Ramban #1 (citing Midrash Tanchuma), Targum Onkelos and Targum Yonasan: It means that Hashem's Hand is raised to swear by His Throne, that He will personally continue to fight Amalek and to treat him like an enemy forever.
Ramban #3 (According to Kabalah): "The Hand which is on the Throne of Hashem" refers to the Midas ha'Din shel Ma'alah, which denotes Hashem's battle against Amalek in all generations. 4
Seforno: The reason that Moshe prayed to Hashem to be his champion 5 was because Hashem swore to fight Amalek from generation to generation, and Yisrael are obligated to fight His battles in every generation.
See Rashbam's objection to this explanation.
Since "Keis Kah (Kisei) Hashem" is synonymous with the king of Yisrael (See Divrei ha'Yamim 1, 29:23 [Ramban]).
The Gemara states in Sanhedrin (20b) that 'Yisrael were commanded three Mitzvos upon entering Eretz Yisrael - to appoint a king, to wipe out Amalek and to build the Beis-ha'Mikdash (Seforno). Refer also to 17:16:1:3**.
Why did Hashem spell the words "Keis" without an 'Alef', and why did he say "Kah" and not the full Name Havayah?
Rashi and Ramban (citing Midrash Tanchuma): He swore that neither His Name nor His Throne will be complete until the Name of Amalek is erased. 1
What does "mi'Dor Dor" mean?
Ramban: That every king of Yisrael is obligated to take the battle against Amalek.
Targum Onkelos: From the generations of the world.
Targum Yonasan: Hashem swore that He would destroy them from three generations: from this world, from the generation of Mashi'ach 2 and from Olam ha'Ba (after Techi'as ha'Meisim).
Why is Amalek punished more than all the other nations who attacked Yisrael?
Ramban: Because all the other nations stood in awe of Hashem when they heard about the miracles of K'ri'as Yam-Suf. 1