What was Reuven's motive in stopping his brothers from killing Yosef?
Seforno: He prevented them from acting suddenly. 1 This leads to deeds that cannot be rectified (e.g., murder).
Ramban (to 37:26): Refer to 37:26:1:3.
Ohr ha'Chayim: People have free choice, are capable of killing someone who is not Chayav Misah. Animals which are under Hashem's jurisdiction, would not kill one who is not Chayav Misah. Reuven therefore saved him from their hands, since killing him would not prove that his dreams were false. 2
Seforno: Also a Tzadik does so sometimes, like Reuven himself did with Bilhah, about which [his father would rebuke him (49:4) for being "hasty like water." He was currently doing Teshuvah, as the commentaries explain, based on Bereishis Rabah (84:19). (EC)
Ohr ha'Chayim: Reuven was sure that animals would not kill one of Yaakov's children. He intended to return Yosef home before he would starve.
Why was it specifically Reuven that stepped in to save Yosef?
Maharal (Chidushei Agados Vol. 4, p. 1, to Makos 10a): Because he was the Bechor, and felt that he was responsible to save Yosef. A Bechor is the initial existence [of a family]. 1
What reward did Reuven receive for this?
Makos 10a: That is why in the Parshah of Arei Miklat, Reuven is mentioned first. 1
See Devarim 4:43. Oznayim la'Torah learns it from the fact that the Torah inserts the word "Bamidbar" both here (37:22) and in Devarim (ibid).
What is the significance of the sequence in these Pesukim, "va'Yatzilehu... va'Yomer Lo Nakenu Nafesh. Va'Yomer Aleihem Reuven... Hashlichu Oso El ha'Bor ha'Zeh"?
Malbim: First, Reuven tried to totally save Yosef (from any punishment). The brothers did not consent. Then he said that they should not kill him. Also this they rejected. Finally, he said that they should not overtly kill him, rather, through Gerama (causation).
Ha'amek Davar: Only in his second statement does Reuven address his brothers, "va'Yomer Aleihem." The first time, Reuven cried out to himself - we cannot kill him! Then he spoke gently but forcefully, and explained why they must consent.