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.
Yes you should capitalize Summer and all other seasons, it's Grammer101
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.
The correct way to capitalize the title "How I Spent My Summer Vacation" is to capitalize the first letter of each word.
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).
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.
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.
Yes, you need to capitalize them.
You should capitalize it if it is a brand name.
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.