Well, mostly because it's a holiday and holidays are capitalized. Not to be rude, but most people learn this in like 4th grade or so.
Christmas should always be capitalized. The merry part needn't always be, but if all your are saying is "Merry Christmas" then the merry gets a capital M because it starts a sentence.
Yes, you always capitalize "Cuban."
The spelling is California, and it is a US state.
Yes, you should capitalize "Merry Christmas" and "Happy New Year" in a letter as they are greetings and proper nouns.
No.
Yes it does.
Yes, you should capitalize the C in Century.
Yes, always.
No you do not.
Yes, you capitalize Spanish because it is a language and culture.
Yes, you should capitalize the "C" in community when referring to the Duke community as it is a proper noun.
Yes, you always capitalize a day of the week.