No, "mid-nineteenth century" is not capitalized unless it is at the beginning of a sentence.
it is not capitalized because the twentieth century is not cool
no
nope!
Yes.
Yes, all words used in the beginning of the sentence is capitalized.
Yes, the name of a century should be capitalized when referring to a specific century, such as the Twentieth Century or the Twenty-First Century.
In the Chicago Manual of Style the c in century is never capitalized. Reference: http://www.chicagomanualofstyle.org/CMS_FAQ/Capitalization/Capitalization09.html
Yes, "Century" should be capitalized when used in reference to a specific century, such as the 21st Century.
It is capitalized at the beginning of the sentence or when it forms part of the proper noun. Examples: The 2nd century is considered part of the Classical era. The Second Century Initiative is celebrating its anniversary.
Yes, Puritanism is capitalized because it refers to a specific religious movement and ideology that originated in 16th-century England.
Only if it's being used as an adjective. e.g. nineteenth-century writers.
It can be. If you are talking about one century as a length of time, ie. one hundred years. It is not. One hundred years ago... One century ago... It's not capitalized. If you are talking about life in the 14th Century, then it is capitalized because you are speaking of a specific set of one hundred years.