1)
Is this Chizkiyah ha'Melech?
1.
Radak, Malbim: Some say that is. We are not forced to say so. In any case, his fathers were great, therefore they are mentioned.
2)
Did he prophesize in all the days of Yoshiyah?
1.
Radak, Malbim: It was before Yoshiyah and Yehudah repented from their evil. Yalkut Shimoni 566 - also Yirmeyahu and Chuldah prophesized then. Yirmeyahu prophesized in the markets, Tzefanyah in the Batei Kenesiyos, and Chuldah prophesized to the women.
