What is the connection between this Pasuk - "Kein Ta'aseh le'Shorcha" and "B'chor Banecha Titen Li" (in the previous one) ?
Rashbam: They are all to do with giving the first (Reishis) to Hashem.
Hadar Zekenim: It teaches us that, if one givs the B'chor ox to Hashem, the rest will be blessed, just like regarding children.
Bechoros 26b: Adding to the Mitzvah of giving one's B'chor to the Kohen, the Torah is obligating a Yisrael to look after a firstborn calf - fifty days, more than after a firstborn lamb or kid-goat -thirty days. 1
Bechoros, 56a: The Torah is comparing B'chor Beheimah to B'chor Adam, to exempt an animal that is purchased or given as a Matanah from the Din of Ma'aser Beheimah - just as purchasing and giving as a Matanah are not applicable to B'chor Adam. 2
Mechilta: It teaches us that, like Bhor Adam, one may give a B'chor Beheimah to any Kohen that one wishes, wherever he lives.
Nowadays, when there is no Beis-Hamikdash, See Torah Temimah, note 220. Refer also to 13:2:1.1:2.
Bechoros (Ibid.): This is confined however, to where, when he purchased the animal, it was otherwise fit to be Ma'asered ("Kein Ta'aseh" - 'be'Sha'as Asiyah'), but not where he purchases ten fetuses, which must all enter the pen to be Ma'asered, even though they were purchased - See Torah Temimah, note 222.
What are the connotations of "Shiv'as Yamim Tih'yeh Im Imo"?
Rashi: It is a prohibition against the Kohen offering up a B'chor before the eighth day. 1
Targum Yonasan: It impies that for seven days it shall nurse from its mother.
Hadar Zekenim: 'Even though I told you to give Me the B'chor promptly, do not rush to give it within seven days] - since it came from a filthy place'.
What are the connotations of "ba'Yom ha'Shemini Titno Li"?
Rashi: What the Pasuk means is that from the eighth day and onwards, the Kohen is permitted to sacrifice it. 1
Seforno: Refer to 22:28:4:2.
Targum Onkelos and Targum Yonasan: It means that one should designate it on the eighth day (to go on the Mizbe'ach).
Chulin, 81a: It means that someone who brings a Mechusar Z'man on the Mizbe'ach is not subject to Malkos, since "Titno Li" is an Asei, and is therefore not subject to Malkos. 2
Which it learns via a Gezeirah-Shavah "Shemini" "Shemini" (in connection with Korbanos), where the Torah writes in Vayikra 22:27 - "umi'Yom ha'Shemini va'Hal'ah Yeratzeh" (Rashi). Torah Temimah: And we learn that other Korbanos may be brought on the eighth day itself from B'chor.
Based on the principle - 'La'av ha'Ba mi'Kelal Asei, Asei' - a La'av that is based on an Asei is an Asei. See also Torah Temimah 127.