Why did the kid-goat play such a prominent role in this episode?
Rashi: Yehudah was being punished here Midah k'Neged Midah - He tricked his father by slaughtering a kid-goat and dipping Yosef's robe in its blood, so he was tricked in connection with a kid-goat.
Why did Yehudah say "I sent this kid"?
Ha'Emek Davar: I sent a special kid, for I recognize that she is esteemed. Also the people realized it was not harlotry. This is why later, Yehudah knew that she is pregnant from him, and did not suspect that she had Zenus also with others.