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.
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).
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."
Yes, you should capitalize "About" in a title.
Yes, you should capitalize the C in Century.
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.
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 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.
You should capitalize Colosseum when referring to the specific ancient Roman amphitheater in Rome.