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.
Yes. It is a noun referring to an organization. No, because military is a general descriptor. Take your pick.
Yes, you should capitalize "The Rotunda" as it is a specific place or location.
You should capitalize Colosseum when referring to the specific ancient Roman amphitheater in Rome.
'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 correct to capitalize "Should" if it is used in a title as it is a noun in this context.