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Amalekites and magic:
The Amalekites' ability to change into animals is brought by Rashi on Shmuel 1:15:3 and the Rid on the same verse, and mentioned as a midrash in the Kitzur Ba'al Haturim's commentary on Shemot 22:17. Their ability to use magic to flee from danger is brought in the longer commentary of Ba'al Haturim on Shemot 17:13 as something that &...
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Prof. Yehudah Elitzur wrote an essay on the subject. In his view, the Amalekite wasn't the son of a convert or even a ger toshav, but an actual 100% Amalekite. What was he doing there? He was a spy. The Amalekites would send off Amalekite youths to collect information in preparation for the many raids the Amalekites did (consider how they first met Yisrael - ...
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According to this answer, it's possible that Agag's mother was not an Amalekite, and so was left alive.
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Ralbag in his commentary addresses all of your points:
The amalekite was simply obeying the King's command. Which law did he breach to warrant the death penalty?
יש לתמוה מדוע צוה דוד להורגו והנה עשה זה ברצון שאול ולבקשתו אלא שהתשובה בזה איננה ממה שיקשה וזה שהאומר לחבירו חבול בי וחבל בו חייב כמו שהתבאר בגמרא כל שכן בענין המיתה וכל שכן בהמתת המלך ואין מצות ...
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