On 56a the gemara brings a tosefta with 6 things that Chizkiyahu did. The chachamin agreed with 3 and disagreed on three.
Who are these chachamin? in the days of Chizkiyahu or Tanaim? If it was in the days of Chizkiyahu how does the Gemara know this? It doesn't bring any pesukim that someone disagreed with the king. If it is the tanaim that disagree with Chizkiyahu isn't this unusual for them to disagree with a righteous king without any proof from the Tanach?
One of the 6 is that he declared an Adar II when it was already Nisan. But declaring an Adar II can only be done by the Sanhedrin? Rambam in perek 4 of kiddush hachodesh goes through the details of what is necessary to declare a leapyear. In particular in needs to be done even bidieved with the permission of the head (nasi) of the Sanhedrin. Lechathila he describes all sorts of procedures that require discussion within the Sanhedrin.
Another one is that he diverted the waters of the Gichon. What is wrong with that? Isn't one supposed to do whatever is necessary and not rely on miracles? Because Isaiah said that the city would be saved does that mean that Chizikayu should not prepare for war? In addition Isaiah's warning came later. Similarly with taking the gold off the Temple doors.
(a) Good question. Judging from the wording used by Chazal, it is certainly possible that these Chachamim did not live in the time of Chizkiyahu. In the discussion of Chizkiyah, they use the terms "Hodu Lo" or "Lo Hodu Lo," and not the terminology that the Mishnah uses ("Michu b'Yado" and "Lo Michu b'Yado") or that the Beraisa uses ("b'Ratzon Chachamim" and "she'Lo b'Ratzon Chachamim"). It could be that the Chachamim in his own time were either subordinate to him, or accepted what he did without protest or expressing disapproval since, as you mention, he was the king. However, from the context it seems that it is referring to Chachamim from his times, that did not agree with him.
Regarding the lack of sources in the verses, it seems that since we find that Chizkiyahu was punished (such as being attacked by Sancheriv), that indicates that there was some sort of iniquity in his hands. The Chachamim searched his deeds and found only six things which might have been improper, three of which they approved of, leaving only three which they did not approve of.
Regarding disagreeing with a righteous king, we know that if a person's teacher is erring in a Devar Halachah, "Ein Chachmah v'Ein Tevunah v'Ein Etzah l'Neged Hash-m" -- nothing stands in the way of the honor of Hash-m, and the student must rebuke his Rebbi (Berachos 19b).
(b) The kings of the dynasty of David were the heads of the Sanhedrin (Sanhedrin 19a, from the verse "Dino l'Boker Mishpat"), and thus Chizkiyahu was a Nasi of the Sanhedrin who decided upon the matter and a majority of the rest of the Sanhedrin of his time followed him. Some of the Chachamim, though, did not approve, as we discussed above.
(c) It is clear from the Sugya that closing off the supply of the waters of Gichon created a great hardship for a large portion of the residents of the city, and for this Chizkiyahu was punished. Regarding your question of a person's requirement to do whatever is necessary, one is not required to do more than is necessary (in this case, to try to defeat the enemies in battle). Cutting off the water supply was beyond the limits of proper preventative action and entered the category of "panic."