Who are Bnei ha'Nevi'im?
Radak citing Targum Yonasan: They are Talmidim of the Nevi'im.
Why does it say "Asher Beis Kel"?
Radak: This is like b'Veis Kel.
How did they know that Eliyahu will be taken today?
Radak (1): Hashem told Eliyahu; he did not reveal it to Elisha. Also Elisha knew via Nevu'ah - "Gam Ani Yadati Hecheshu" (verse 3). It was told to the other Nevi'im, and all the more so to [Elisha]!
Malbim: Mekubalim say that an Ish ha'Elokim amidst his Talmidim is like the heart that gives life to the body. Limbs closer to the heart receive more life and influence. When he departs, the limbs further away sense the departure before the close limbs. Therefore, Bnei ha'Nevi'im sensed first that their heart and Nefesh will leave.
Why did they call him "Adonecha", and not Adoneinu?
Rashi (from Tosefta Sotah 12:5): This teaches that they were equal to Eliyahu 1 .
Tosefta Sotah 12:5: They were his colleagues. After Eliyahu was taken, Ru'ach ha'Kodesh left them, and they called Elisha their Adon.
What is the significance whether or not Elisha knows that Eliyahu will be taken today?
Malbim #1: An Ish ha'Elokim amidst his Talmidim is like their heart (refer to 2:3:1:2). If also Elisha, who is close to him, feels that he will be taken, this is a Siman that it will happen very soon.
What is the meaning of "Hecheshu"?
Radak citing Targum Yonasan: This is a command (be quiet). It is as if there is a Patach under the Hei.
Radak citing his father: They were quiet.
Malbim: You need not pray. Eliyahu will not be taken due to sins of the generation, rather, because he reached his perfection (refer to 2:3:3:2).