What does this Sefer discuss?
Radak: It discusses the evil that Nebuchadnetzar did to Yisrael more than all nations, that the evil that Paras and Madai will do to his seed in the end. Chabakuk begins to complain to Hashem about Resha'im who harm Yisrael, and succeed in their ways. "Tzadik" (4) refers to those in his generation, or afterwards (he prophesized what Nebuchadnetzar will do).
Malbim: First he complains about Bavel's success. He receives an answer, and is informed of Bavel's fall and destruction in the days of Belshatzar.
What is "ha'Masa Asher Chazah"?
Rashi: It is the Nevu'ah that he bore and received with Ru'ach ha'Kodesh.
When did Chabakuk live, and what was his family?
Radak: The Sefer does not tell. It seems to me that Nebuchadnetzar began reigning in his generation, for he says to people of his Dor "Ki Fo'al Po'el bi'Ymeichem" (5).
Radak citing Seder Olam (20): Yo'el, Nachum and Chabakuk were in the days of Menasheh. Because Menasheh was improper, it does not say that their prophesies were in his days.
Zohar (introduction, 7b): He was the son of the Shunamis. He is called Chabakuk, for two were Chavukim (embraced with) him ? his mother ("At Chovekes Ben" - Melachim II, 4:16), and to revive him, Elisha put his mouth on his mouth?