Not sure about the Putin part. Putin obviously wanted Trump, which was obvious from Kremlin-sponsored media, for instance. He probably does have some kind of compromising tape.
I think the last sentence is a stretch. Putin hated Clinton, because he sees NATO as a foreign aggressor, and Clinton would have promoted NATO offensive capabilities. Trump was more flexible on NATO, and thus was preferable, without any “compromising tape.”
(Besides, Trump appears totally shameless. His wife and kids know exactly “who he is”, from the standpoint of moral behavior. I can’t imagine anything that would embarrass him enough, or anything his family hasn’t at least imagined could be true, that could change his behavior. You can’t blackmail somebody who is unconcerned about how he is percieved by others.)
Tillerson went to Moscow and laughed at American journalists at press conferences.
Being the CEO of the largest company in an industry that the media hates, I suspect Tillerson has a view of the media that is similar to Trump’s, but independently derived.