Yes, they are police; Chin Ho said as much to his cousin, and Danno said as much to the lady at the concession stand. Not to mention that McGarrett was able to administer Kono's oath as a new graduate of the Police Academy, something he couldn't do unless he were a police officer. They're a state police unit rather than local police, but they're still police.
And it doesn't matter how much authority the governor gives them, they'd still have to follow proper evidence handling and other procedures if they had any hope of getting convictions in a court of law. The protection against illegal search and seizure is a constitutional right, not a local statute; no state governor could have the authority to subvert the Constitution of the United States, and any governor who tried would be facing impeachment once it came out.
And it doesn't matter how much authority the governor gives them, they'd still have to follow proper evidence handling and other procedures if they had any hope of getting convictions in a court of law. The protection against illegal search and seizure is a constitutional right, not a local statute; no state governor could have the authority to subvert the Constitution of the United States, and any governor who tried would be facing impeachment once it came out.