To whom is this Pasuk referring?
Rashi #1 (in the original manmuscript): It is only those who adhere to the B'ris and who keep the Mitzvos who are fit to Pasken. 1
Rashi #2 (in Shabbos, 114a): If the Kohanim were chosen to teach Torah to Yisrael. It is a sign that they are "Zerizin" - 'agile' (in their learning) and that they are experts across the board.
Ramban (in Pasuk 8): The first half of the Pasuk ("Yoru Mishpatecha le'Ya'akov... ") is referring to the entire tribe, 2 and the second half, to the Kohanim exclusively.
Yoma, 26a: It is only in the triobes of Levi and Yisachar that Talmidei-Chachamim who are ab leto Pasken le'Halachah are to be found, Levi'im from the current Pasuk, Yisachar, because the the Pasuk writes in Divrei Hayamim 1, 12:33 "u'mi'Benei Yisachar Yod'ei Binah le'Itim ... ". See Torah Temimah, note 33 for additional references..
What are the implications of "Yoru Mishpatecha le'Ya'akov"? How did they implement it?
Rashi (in Mal'achi, 2:7): It implies that the Kohanim have been handed the tradition of Torah and that they are therefore the ones to whom one turns for a P'sak Halachah - as the Navi writes in Mal'achi (Ibid.) "Ki Sifsei Kohen Yishmru Da'as ve'Torah Yevakshu mi'Piyhu".
Rashi #1 (in Shabbos, 114b): It is the source of the principle "Kohanim Zerizim hein" - 'they are experts in all (matters concerning Torah).
Ba'al ha'Turim: It teaches us that the Kohanim should scatter among all the tribes in order to teach them Torah.
Yoma, 27a: One will not find a Talmid-Chacham who is a Posek who does belong either to the tribe of Levi, 1 as taught in thee current Pasuk, or to the tribe of Yisachar, as the Pasuk writes in Divcrei Hayamim 1, 12:33 "u'mi'Benei Yisachar Yod'ei Binah la'Itim, Lada'as ... ".
See Torah Temimah in Vay'chi Bereishis, 29:10 DH 've'Ayein be'Yoma, 26a, as to why the Gemara does not incorporate Yehudah in the current distinction.
What is the connection between this Pasuk and the previous Pesukim?
Rashi: Following the dedication of the Levi'im by the Eigel, they are worthy of becoming the teachers of Yisrael.
Rashi #2 (in the original manuscript): The Torah juxtaposes "Yoru Mishpatecha le'Ya'akov" to 'u'Verisxcha Yintzoru" to teach us that only someone who 'guards his B'ris' is fit to Pasken.
Ramban (in Pasuk 8, citing Targum Onkelos) and Targum Yonasan: Refer to 33:8:5:1. Having discussed the dedication, first of the Kohanim (Aharon) and then of the Levi'iim, who proved themselves by the Eigel, in Egypt (refer to 33:9:3:2) and in the desert, 1 the Torah concludes that this is why they merited to become the teachers of Yisrael.
Seforno: Since Levi is a righteous tribe, grant them the Chein and the wisdom to teach Your people your judgments'. 2
Oznyim la'Torah: The juxtaposition of "Yoru Mishpatecha le'Ya'akov" to "ha'Omer le'Aviv u'le'Imo Lo Re'isiv ... ' hints at the Mishnah in Avos 'Hevey Goleh li'Mekom Torah' - that it is only someonme who forgets his family and leves home to learn Torah for long periods who merits becoming a genuine Posek.
Seforno: As the Gemara states in Mo'ed Katan, 17a - 'If the Rav is like an angel of G-d, seek Torah from his mouth!' Mishnas R. Aharon (4 p.21) - they merited "Yoru Mishpatecha le'Yisrael" because at a time of laxness, not only did they not bow to the Eigel, but they answered Moshe's call "Mi la'Hashem Eilai!"
What is the difference between "Mishpatecha" and "Torascha"?
Toldos Yitzchak: "Mishpatecha" refers to monetary laws 'Bein Adam la'Chavero', whereas "Torascha" refers to Mitzvos 'Bein Adam la'Makom'. 1
Oznayim la'Torah: "Mishpatecha le'Ya'akov" refers to the common people, who have financial issues with one another, for which they turn to the Kohanim to adjudicate; whereas ''Torasecha le'Yisrael" refers to the Talmidei-Chachamim who tend to forego their finn=ancial claims against one another, and who turn to the KOhanim to teach them Hashm's Torah.
Toldos Yitzchak:And the Torah refers to the nation regarding the latter as Yisrael" which is a higher level than "Ya'akov". This is not clear however, since the Kohanim are ob;igated to teach both groups of Mitzvos to all of Yisrael! (PF)
What is "Yasimu Ketorah be'Apecha" referring to?
Targum Onkelos: It is referring to the Ketores, which was brought every morning and every afternoon.
Targum Yonasan: It is referring to the Ketores that the Kohen brings (Rosh
See Korach, Bamidbar, 17:11. Targum Yonasan implies that this does not refer to the plague after Korach's rebellion. This is like Rashi in Bamidbar 17:11 - that the angel of death taught Moshe that the Ketores stops plagues - and [Kohanim] may burn the Ketores without explicit permission from Hashem). However, there (in 17:13) Rashi implies that Hashem specifically authorized it. (PF).
Which is why, once a Kohen burned the Kerores, they did not allow him to burn it again, as the Mishnah states in Yoma, 2:4..
Refer to answer #2 and note.
What is the Definition of "Kalil"?
Rashi and Targum Yonasan: "Kalil" refers to the Korban Olah - which is completely (Kulo) burned. 1
Oznayim la'Torah: And which was brought on the MIzbe'ach only after the Ketores was burned, in keeping with the daily order of the Korbanos of Aba Sh'aul which we read each morning in Shachris.
Why does the Torah not mention the Mizbe'ach in connection with the Ketores - like it does by the Olah?
Oznayim la'Torah: This supports the ruling of the Rambam, that if , when the Kohen comes to burn the Ketores, he does not find the Mizbe'ach ha'Zahav in the Heichal, he places the pan on the floor and burns it there. 1
See Oznayim la'Torah, DH 'Yasimu Ketorah be'Pecha ve'Chalil al Mizbechecha'.