Why did Moshe teach Yisrael about the T'reifos via Aharon and his sons?
Rashi: It was a reward for remaining silent and accepting Hashem's decree with love when Nadav and Avihu died. 1
Ramban: Because most of the issues regarding the Kashrus of animals concerns the Kohanim in particular. They had to beware against touching Tum'ah, since they needed to come to the Beis-ha'Mikdash (to bring Korbanos) and to eat Kodshim. Also if a Yisrael transgressed be'Shogeg, he had to bring a Korban - which the Kohanim would then sacrifice. Moreover, they needed to teach the people the difference between Tamei and Tahor, with regard to eating and the Dinim of Tum'ah, so that they should not render Tamei the Mishkan and its Kodshim. 2
Why does the Torah write "Zos ha'Chayah" rather than 'Zos ha'Behemah'?
Rashi: This hints at the fact that Hashem separated Yisrael from Tum'ah and issued them with Mitzvos, 1 because, since they cleave to Hashem they deserve to live.
Which He did not do with the Nochrim. This can be compared to a doctor who came to visit two patients. When he instructed the family of one of them to allow him to eat whatever he fancied, and the other he placed upon a strict diet, he explained that the former was doomed to die, whereas the latter was able to live, and by what he ate, he would fully recover (Rashi, citing Midrash Tanchuma).
What are the implications of the word "Zos"?
Torah Temimah (citing Chulin, 42a): "Zos ha'Chayah asher Tocheilu" implies that one may only eat from an animal that can live - to preclude a Tereifah, which cannot live. 3
And why does the Pasuk write both "Zos ha'Chayah" and "mi'Kol ha'Beheimah"?
Why does the Pasuk add "mi'Kol ha'Beheimah asher al ha'Aretz"?
Ramban #1: It simply means from all the animals on the earth (in the world).
Ramban #2: It comes to exclude the sea-animals, which do not share the same signs of Kashrus as their land counterparts. 1
See also Ba'al ha'Turim.
Why does the Torah insert the Parshah of Kosher and non-Kosher animals in Seifer Toras Kohanim (Vayikra)?
Ramban: Refer to 11:2:1:2.
Seforno: Refer to Yisro 20:21:5:4*. With his Tefilos, Moshe succeeded in bringing back the Shechinah in a limited manner, via the Mishkan and on condition that Yisrael live up to it by maintaining a high standard of spirituality, incorporating avoiding Tum'ah 1 and abstaining from sins that the Torah considers abominable. 2