Why does the Torah begin a new chapter with a 'Vav'?
Oznayim la'Torah: Having already taught some Mishpatim at Marah, 3 the Torah is now coming to add the rest.
Because the 'Vav' in "ve'Eileh" signifies that what follows is connected to what was written before it. (See Sifsei Chachamim).
Oznayim la'Torah: And the Torah sees fit to do so specifically here in connection with Mishpatim, since we might otherwise have thought that the Dinim were taught by Moshe, just as the other nations, who were also commanded Dinim, the dettails of which their elders had to work out for themselves. Therefore the Torah teaches us that, in contrast, Yisrael were taught at Har Sinai every detail contained in 'MIshpatum' (even that a Tam pays half and a Mu'ad, in full)
Why does the Torah juxtapose the Parshah of Dinim (Mishpatim) to the previous Parshah?
Rashi: It juxtaposes it to the Parshah of Mizbe'ach to teach us that the Sanhedrin must sit near the Mizbe'ach. 1
Ramban and Seforno: The Torah inserts Mishpatim next to "Lo Sachmod", which ends with "Kol asher le'Re'echa" to teach us what is considered "Kol asher le'Re'echa". 2
Moshav Zekenim (20:23) #1: One should be deliberate in judgment, just as one may not ascend the Mizbe'ach quickly, on steps ("ve'Lo Sa'aleh be'Ma'alos al Mizb'chi").
Moshav Zekenim (20:23) #2: This warns a judge not to [appear to] step on the heads of the Am Kodesh (Yisrael).
Hadar Zekenim: If one will not punish murderers, adulterers and thieves, he will be sold due to their theft - "v'Nimkar bi'Gneivaso."
The Lishkas ha'Gazis, where the Sanhedrin convened, was situated in a room adjoining the Azarah, close to the Mizbe'ach.
After elaborating on the Isur of Avodah-Zarah at the end of Sh'mos 20:19 & 20 (Ramban). In fact, the Torah also inserts details concerning Avodah-Zarah, (Sh'mos 22:19), Kibud Av va'Eim (Sh'mos 21:15, 17), murder (Sh'mos 21:12, 14) and adultery (Sh'mos 22:18), all in the current Parshah (Ramban).
Why does the Torah interrupt between the Parshah of the Mizbe'ach and that of the Mishkan with Parshas Mishpatim?
Why does the Torah interrupt between the appointment of Shoftim in Yisro and that of Mishpatim with Matan Torah?
Oznayim la'Torah (citing Midrash Rabah): So that there should be Dinim before Matan Torah and Dinim after it.
What are the connotations of "Asher Tasim Lifneihem"?
Rashi #1: Hakadosh-Baruch-Hu said to Moshe 'Don't think that you can simply teach them the subject-matter two or three times until they have grasped it, without teaching them the reasons and the explanations!' Therefore the Torah says 'Asher Tasim Lifneihem' 1 - Like a set table, ready to partake of the food. 2
Seforno: The Lashon "Asher Tasim Lifneihem" implies that, as opposed to the warnings in Yisro, the Pesukim in Mishpatim are not Mitzvos Asei or Lo Sa'aseh per se, only when the need arises, this is how the Dayanim should judge them.
Moshav Zekenim: This refers to the Kelim of judges - a staff and whip to force people to comply with the rulings of Beis-Din.
Why does the Torah add the word "Lifneihem"?
Rashi and Ramban #1 (citing the Tanchuma and Gitin, 88b): "[Come for judgment] in front of them" but not in front of Nochrim 1 - even if one knows for sure that they issue the same ruling as Din Torah. 2
Ramban #2 (Ibid.) "Before them" but not before laymen. 3
Rashi #2 (in Gitin, 88b): It actually refers to the seventy Zekeinim who accompanied him on to Har Sinai before Matan Torah. 4
Da'as Zekenim: These laws are only for Yisrael, for they live through them. They would not help for Nochrim, so He did not command them.
Kidushin, 35a: In order to equate women with menregarding all Dinim in the Torah. 5
Ramban: Otherwise, the Torah ought to have written 'Asher Tasim lahem' - similar to the expression that the Torah uses in Sh'mos 15:25 and in many other places. Oznayim la'Torah: Although this seems to be a prohibition addressed to the litigants against bringing their cases before Nochri courts, it may be a warning to Moshe,against bringing in expert Nochrri judges and teaching them the Mishpatim of the Torah - bearing in mind that Yisrael had been slaves for hundred of years, and were therefore totally ignorant concerning all judiacial issues.
Because someone who brings a Din Torah before Nochrim desecrates the Name of Hashem, and gives honor to the name of Avodah-Zarah, to praise it, as the Torah indicates in Devarim 32:31 (Rashi).
See later in Pasuk 6, which clearly states that Dinim lie exclusively in the realm of expert Dayanim (See Ramban who cites many other sources for this). And this applies even if the laymen concerned are conversant with Din Torah, only they are not authorized to Pasken - since the issue at stake is one of forcing the litigants to comply with the Dayan's P'sak (See Ramban, who elaborates further). See also Torah Temimah, note 11.
See later, Sh'mos 24:1 (Rashi).
Moshav Zekenim: Why is this needed? Since we need a Drashah to disqualify women from testimony, surely they must be punishable. If not, their testimony could not be Huzam! This requires investigation. (Perhaps the Drashah is needed for testimony to which "Ka'asher Zamam" does not apply, e.g. about lineage, according to one answer in Tosfos Makos 2a, or Kidush ha'Chodesh. - PF).
QUESTIONS ON RASHI
Rashi writes that had it said 'Eleh', this is Posel (disconnects it from what is written earlier), and 'v'Eleh' adds to what came before. There are exceptions to this! And here, in any case it could not be Posel! "Va'Yered Hashem Al Har Sinai" - they were said at Sinai!
Moshav Zekenim #1: Whenever it can be Posel, 'Eleh' is Posel. Here, it could not be Posel, for it says "va'Yered
Moshav Zekenim: Why is this Posel? Hashem said everything! (Ohel Yakov (Reish Devarim, citing the Vilna Gaon) - Hashem said the first four Chumashim through Moshe, but Sefer Devarim Moshe said not at the time of prophecy, like other Nevi'im. Maharal (Tiferes Yisrael 43) - Moshe himself said Sefer Devarim, for it is a Perush for the sake of the receivers of Torah!)
Moshav Zekenim: This is difficult. Had it said 'Eleh', would all not be fixed? Hashem told Moshe the previous laws, and also these!
Rashi writes that Parshas Mishpatim was put next to the previous Parshah to teach us that the Sanhedrin must sit next to the Mizbe'ach. We already know this from "v'Alisa El ha'Makom
Bartenura: "V'Alisa El ha'Makom" teaches only that the Sanhedrin must be in the Mikdash. It does not teach that they must sit next to the Mizbe'ach.
Moshav Zekenim: "El ha'Makom" teaches that the place is essential (Zaken Mamrei applies only if the Sanhedrin is in the Mikdash).
Rashi writes that "Lifneihem" excludes Nochrim. Why is this needed? Lifneihem excludes even converts (Kidushin 76b) and commoners!
Riva: Converts are excluded only when there are Yisraelim who know to judge. Shemayah and Avtalyon were converts, and the judged Yisrael, for there was no Yisrael as esteemed as them!
Moshav Zekenim #1: Lifneihem disqualifies the verdict even b'Di'eved. The verdict of converts and commoners is valid b'Di'eved.
Moshav Zekenim #2: We exclude commoners from "Elohim", and not from Lifneihem. Even though "Lo ch'Tzurenu Tzuram" (Devarim 32:30) forbids going to Nochri courts, we need Lifneihem to forbid even if they judge this case like Torah law. It is permitted only if Yisraelim judged, and the Nochrim say 'fulfill what Yisrael tell you.'
Rashi writes that "Lifneihem" excludes Nochrim. If so, we should infer that women can be judges. Bava Kama 15a infers that men and woman are equated for all Dinim! And it says that Devorah judged!
Moshav Zekenim: They are equated only for what is written explicitly about men. Devorah taught the laws, through Nevu'ah. 1
Moshav Zekenim: We conclude that women can judge.
Why is this permitted? "Lo va'Shamayim Hi" (Devarim 30:12)! Perhaps this forbids only Chidushim (Temurah 16a). She taught only laws known to Chachamim of her generation. (PF) Tosfos (Bava Kama 15a DH Asher) - both parties can agree to let a Pasul judge their case. People accepted her for a judge due to Shechinah.