Wat are the implications of "Shor ... ki Yivaled"?
Rashi #1 and Targum Yonasan: It implies that only an animal that is born naturally can be brought as a Korban, but not one that is born by caesarian section.
See K'li Yakar.
Why does the Pasuk see fit to specify the three animals individually?
A baby born from two different species.
A sheep that gives birth to a kid-goat, or a goat that gives birth to a lamb.
See Targum Yonasan.
Targum Yonasan: Because nowadays, when there are no Korbanos, we should remember that our Tefilos, which come in place of the Korbanos, are answered on the merit of the Avos (See Na'ar Yonasan).
Why is the reason for the prohibition to Shecht a Korban before the eighth day?
Ba'al ha'Turim: Because on the first day it conveys the impression that one is Shechting to the heaven and earth, on the second day, to the sky, on the third day, to the sea and dry land, on the fourth, to the celestial luminaries, on the fifth day, to the rodents, on the sixth day, to man. One should therefore wait a full seven days, when it will be clear that he is Shechting to the G-d who created the world in six days and rested on the seventh.
Targum Yonasan: To ensure that it is not a Nefel (a stillborn baby).
What is the connection between this Parshah and the previous one?
Seforno: Having discussed the blemishes that disqualify an animal from the Mizbe'ach - even though sometimes a blemished animal is worth more than its complete counterpart, and even though one would not hesitate to bring it before one's king 1 - the Torah now discusses perfection in time, forbidding the bringing of animals before the eighth day as Korbanos, Shechting a mother and her baby on the same day and disqualifying a Korban even with the thought of eating it after its allotted time.
Seforno: Because Hashem, who is perfect, demands physical perfection regarding the animals that are brought before Him and as close to spiritual perfection as is possible regarding those bringing the Korban.