If it's a key word in a title, or the first word of a title or part of a name of something, capitalize. Otherwise, don't.
If you are using it in a title.......... How I Spent My Summer Vacation.
Yes, "Summer" should be capitalized when written as "Summer 2007" as it is part of the name of the season and is used as a proper noun in this context.
No.
Summer should be capitalized when used as a proper noun, such as "I love spending time at the beach in Summer." However, it should not be capitalized when used as a common noun, such as "I enjoy the warm weather in the summer."
No, unless it is at the start of an sentence (In this case you capitalize only 'summer') or part of a title/name (Eg. Google Summer Internship).
Yes, you should capitalize "About" in a title.
Wyoming should be capitalized at the beginning of a sentence or when referring to the state specifically as in "I visited Wyoming last summer."
Yes, you should capitalize the C in Century.
Yes.
YOU CAPITALIZE THINGS LIKE SPRING, SUMMER, AND WINTER WHEN YOU USE PERSONIFICATION.FOR EXAMPLE, As I was walking down the street, Winter came and blew away my hat.
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.