no
New York City should be capitalized with each word starting with a capital letter.
Yes, all three words in New York City are capitalized. Unless, of course, you are referring to some other city in New York State besides New York City, for example, "The only New York city I haven't been to is Buffalo."
Yes.
Yes, you should capitalize the word "Northern" when referring to a specific region like Northern New York.
New York City is the largest city in New York State.
New York City is a city within the State of New York. There are no states "in" New York City.
New York, as it contains New York City.
New York city is a city in New York state.
New York City is known as "The Big Apple".
New York City.
New York City! New York City!
Los Angeles.