Why did the brothers take specifically a goat to dip Yosef's robe in its blood?
Rashi and Targum Yonasan: Because the blood of a goat resembles that of a human-being. 1
Chizkuni: When they told Nevuzaradan that Zecharyah's blood was of Korbanos, they slaughtered Korbanos and it did not resemble their blood (Gitin 57b)! That is because [the goat's blood] was mixed with others' (cattle and sheep. Riva explained that other bloods were mixed with Zecharyah's). Maharam Shif (57b) dismisses the question. When they are next to each other, one can distinguish them. Further, Zecharyah's blood was boiling, unlike blood of Korbanos!
Why does it say that they took the robe before they slaughtered the goat?
Ha'Emek Davar: Until now hey left it on the ground and did not use it, lest it recall their sin. Now they took it to be able to dip it in the blood immediately after Shechitah, before the blood is absorbed in the ground. 1
One may not slaughter [so that the blood will fall] into a Kli, lest people say that he intends to throw the blood to idolatry (YD 11:3). However, it is permitted if there is earth in the Kli. It is (PF)