What are the connotations of "Ki Yipalei Mimcha"?
Rashi: It means that it is separated (covered or hidden) from you.
Sanhedrin, 87a; "Ki Yipalei" refers to a member of the Sanhedrin (Mufla she'be'Beis-Din), and "mimcha" means that he is capable of arranging a leap-year and fixing the months. 1
Oznayim la'Torah: It refers to a Beis-Din who cannot decide a certain Halachah, due to a difference of opinion within their ranks.
Whereas a Talmid is not subject to the Din of a Zakein Mamrei.
What are the implications of "Davar la'Mishpat"?
Sanhedrin, 87a; "Davar" refers to a Halachah le'Moshe mi'Sinai and "la'Mishpat", to something that is learned from the thirteen principles of R. Yishmael.
Sahedrin, 87a: We learn via a Gezeirah Shavah "Davar" "Davar" from the Par He'elam Davra, in Vayikra, 4:13, that the Zakein Mamrei is only Chayav if he rules concerning a matter that is subject to Kareis be'Meizid and a Chatas, be'Shogeg. 1
See Torah Temimah, note 32.
What are the implications of "Bein Dam le'Dam" "bein Din le'Din" "u'Vein Nega la'Nega", respectively?
Rashi: They imply that the local Beis-Din does not know whether a. the blood of a woman is Tamei or Tahor, 1 b. the accused is guilty or innocent) c. 'a Nega has Tum'as Tzara'as or not'. 2
Ramban: Whether it concerns a. the blood of Nidus, 3 of childbirth or of Zivus; b. cases concerning capital punishment, of monetary issues 4 or of Malkos; c. Tzara'as of people, garments or houses. 5
Nidah, 19a: This Pasuk serves as the source for the fact that not all the blood of a woman is Tamei. See also Torah Temimah, note 34. Targum Yonasan explains "Bein Dam le'Dam" and "u'Vein Nega la'Nega" lik Rashi (in answer #1) and "bein Din le'Din" like the Ramban (in answer a#2).
The lists them in order - from the less difficult to the more difficult. See Oznayim la'Torah.
See Torah Temimah, note 33 & 35.
See Torah Temimah, note 36.
See Ramban who elaborates. See also Torah Temimah, note 37.
What is "Divrei Rivos" referring to?
Rashi, Seforno, Targum Onkelos and Targum Yonasan: It refers to the Chachamim of the town who argue over the matter - some say Tamei and others say Tahor; some declare the accused Chayav, others Patur. 1
Ramban: It incorporates a. giving a Sotah the water to drink; b. breaking the neck of the Eglah Arufah and c. purifying a Metzora.
Sanhedrin, 87a: "Divrei" incorporates Charamim, Arachim and Hekdeishos, and "Rivos" - see answer #2.
Seforno: Even though the Torah appointed Dayanim in every town, they do not have a mandate to resolve doubts that arise in the tradition handed down from Sinai, but are obligated to consult the Beis-Din ha'Gadol, as they must also do if there is no majority opinion in their own Beis-Din.
What are the connotations of "bi'She'arecha"?
Targum Onkelos and Targum Yonasan: It means in your towns - Targum Onkelos, or 'in your Batei-Din - Targum Yonasan.
Sanhedrin, 87a: It incorporates Leket, Shikchah and Pe'ah in the Din of Zakein Mamrei. 1
See Torah Temimah, note 38, who elaborates and clarifies how all the cases can lead ro a Chiyuv Kareis..
What is the meaning of "ve'Kamta ve'Alisa"?
Rashi (In the original manuscript): It is coming to teach us that the fate of the Zakein Mamrei is due to the location
Why does the Torah write "ha'Makom asher Yivchar Hashem ... "?
Rashi (in his initial explanation of the Chumash): It teaches us that the liability of Zakein Mamrei depends on the location. He must rebel against the Beis-Din ha'Gadol in the Lishkas ha'Gazis (But if he found them in the area of Beis Pagi 1 and rebelled against them there, he does not have the Din of a Zakein Mamrei - Sanhedrin, 14b) - because the "Makom" is crucial.
A location that is just inside the walls of Yerushalayim - See Torah Temimah, note 42.
Seeing as the Torah is discussing a Beis-Din who disagree over a certain Halachah (See 17:8:1:3), why is the entire Parshah written in the singular?
Oznayim la'Torah: Because their visit to the Beis-Din ha'Gadol is not in the form of two disputing groups, to find out which of them is right, but as one group, to clarify the Halachah.