I don't understand grammar 'to hate did not back down' and meaning of whole sentence. Is it special style of the news?
President Trump’s favorite news outlet to hate did not back down from the commander-in-tweet’s recent attack on its reporting.
You're parsing it slightly incorrectly. It's not "to hate did not back down", it's two separate phrases.
( President Trump's favorite news outlet to hate ) ( did not back down )
That is, the subject in the sentence is "President Trump's favorite news outlet to hate", which means "The news outlet that President Trump is most fond of hating."
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1Just to state it more explicitly, out of all of the news channels, papers, web sites etc. this particular news “outlet” is the one that President Trump criticizes most often. I have a feeling that the meaning of “the one someone loves to hate” might not be easy to understand for some learners. – ColleenV parted ways♦ Aug 30 '18 at 19:56
It is not a phrase because the sentence should be parsed differently. It goes like this
President Trump’s favorite (news outlet to hate) did not back down from the commander-in-tweet’s recent attack on its reporting.
So to hate follows news outlet and describes what it does and is its attribute. And the news outlet didn't back down from the commander-in-tweet’s recent attack on its reporting.
The sentence core is:
Trump's news outlet did not back down
I think you're confused about how to break that down into parts of speech.
The subject of the sentence is not "President Trump's favorite news outlet".
Rather, it is "President Trump's favorite news outlet to hate".
