Why does the Torah write "Tasim alecha Melech" and not 'Asim alecha Melech'?
Torah Temimah: Because Yisrael did not really need a king - since Hashem is their King, 1 and it is only because Hashem foresaw that they would (reject Him and) ask for a king that He turned the request into a Mitzvah.
As Shmuel told the people in Shmuel 1, 12:13. Refer also to 17:14:2:1 and note 2.
What are the implications of "Som Tasim alecha Melech!"?
Ramban #1 (citing the Sifri): To teach us that if the current king dies it is a Mitzvah to appoint another one in his place. 1
Ramban #2 (citing the Sifri): It mplies "Melech", 've'Lo Malkah' - to disqualify a queen from sitting on the throne.
See Torah Temimah, note 79.
What are the implications of " mi'Kerev Achecha Tasim alecha Melech"?
Ba'al ha'Turim: "mi'Kerev Achecha" has the same Gematriya as 'mi'Sheivet Yehudah'.
Sifri: It implies that the king must be from Eretz Yisrael and not from Chutz la'Aretz. 1
Yevamos, 120a; "Alecha" implies that a Ger who is a Dayan is not permitted to judge a Yisrael but he is permitted to judge his fellow-Ger. 2
Kidushin, 76b #1: It means that his mother is a Bas Yisrael 3 (even though his father is a Ger). 4
Kidushin, 76b #2: It implies that every important appointment 5 should be from you - someone whose mother is a Bas-Yisrael - and not a Ger. 6
Bava Kama, 88a: It precludes anyone who does not have a Yichus - such as an Eved Cana'ani - from becoming king. 7
Why does the Torah add the (otherwise superfluous) words "mi'Kerev Achecha Tasim alecha Melech"?
Sanhedrin, 22a:
Kidushin, 32b: See answer #1. From the king's perspective, it prohibits him from foregoing his Kavod - 'Melech she'Machal al Kevodo, Ein Kevodo Machul'. 1
See Torah Temmah, note 77.
What are the implications of "asher Yivchar Hashem Elokecha bo"?
Ramban: It implies that the appointment must be made through a Navi or via the Urim ve'Tumim. 1
Targum Yonasan: It implies that, before choosing a king, one should consult Hashem. 2
Why does the Torah repeat "Tasim alecha Melech"?
Kidushin 32b: To teach us that a king cannot forego his honor 1 - 'she'Tehei Eimaso alecha'.
Sanhedrin, 22a and Sifri #1: To teach us that one be constantly awe-stricken before the king, and that consequently, one may not ride on his horse, sit on his throne, use his scepter see him whilst he is having a hair-cut, when he is naked or whilst he is in the bath-house. 2
Sifri #2: To teach us that it is a Mitzvah to appoint a king. 3
Yerushalmi Kidushin, 4:5: To include all important appointments in the current restriction - such as heads of police, Gaba'ei Tzedakah, the scribes of Beis-Din and those who whip the malcreants. 4
Because his honor, which is a reflection of Kavod Shamayim, is not his to forego.
This is something that distinguished the era of the kings from that of the Shoftim. See Oznayim la'Torah, DH 'Som Tasim' #1.
See Torah Temimah, note 78.
Refer to 17:15:1.1:5.
What are the connotations of "Lo Suchal La'seis alecha Ish Nochri"?
Rashbam: It is a prohibition against appointing a Nochri king to fight their battles - because he will lead them astray to serve Avodah Zarah. 1
Oznayim la'Torah #1: It means, not literally a Nochri, but somewhat who is estranged from the Torah. 2
Oznayim la'Torah #2: It means that, even if a Nochri offers to rule over Yisrael, they should refuse his offer. 3
Neither the Rashbam nor Targum Onkelos, in answer #3, contend with the main point of the question. Presumably, they will both learn like the Ramban in answer #1.
See Oznayim la'Torah, DH 'mi'Kerev Achecha', who proves that Targum Yonasan also interprets "Ish Nochri" in this way.
See Oznayim la'Torah, DH 'Lo Suchal ... '.
Seeing as the king is chosen by a Navi, what does the Torah mean when it writes "Lo Suchal Laseis alecha Ish Nochri"?
Ramban #1: To teach us that, even if, for some reason, Hakadosh-Baruch-Hu declines to answer through Nevi'im, it is forbidden to crown a Nochri king.