No it doesn't require capitalization.
No, but you should capitalize American. If the specific branch of the military is named, it should be capitalized.
Yes, it is a proper noun.
Yes, "Military Services" should be capitalized when referring to the armed forces of a country or any organization related to military activities.
Yes, "Infantryman" should be capitalized as it is a specific job title or role within the military.
Yes, you should capitalize "About" in a title.
Yes, you should capitalize the C in Century.
No, you should not. You should only capitalize a word when it is at the start of a sentence or when it is a proper noun.
You should capitalize Colosseum when referring to the specific ancient Roman amphitheater in Rome.
Yes, you should capitalize "The Rotunda" as it is a specific place or location.
Yes. It is a noun referring to an organization. No, because military is a general descriptor. Take your pick.
'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).
It should not be capitalized.