Yes. It is a noun referring to an organization. No, because military is a general descriptor. Take your pick.
Yes.
No, but you should capitalize American. If the specific branch of the military is named, it should be capitalized.
No it doesn't require capitalization.
'US' must be captialized, but for the 'the' you should only capitalize it if it is at the start of a sentence (Eg: The US is a big country).
Yes, it is a proper noun.
Yes.
Yes. US Government is capitalized.
Yes.
US Military or U.S. Military.
US Border Patrol agent
Only if it is the contraction of United States.
You capitalize all rank that is part of a title, for instance:Private QunicySergeant JonesCaptain WallyIt is not necessary to capitalize rank that is not part of a title, for instance."John wanted to become a sergeant in the US Army."
No it shouldn't be capitalized.