No, except at the beginning of a sentence because it is
n
ot a proper
n
ou
n.
Yes. 'Sun' is its name and you capitalize it just as you capitalize anyone's name.
Yes, you always capitalize "Cuban."
You do capitalize Sun because it's a specific star with a name.
Yes, you capitalize Spanish because it is a language and culture.
Yes, always.
No you do not.
Yes, you always capitalize a day of the week.
Yes, always capitalize a state.
Yes, you should always capitalize a proper adjective.
The proper adjective for the proper noun Shakespeare(always capitalize a proper noun) is Shakespearean (always capitalize a proper adjective).
Always. It is a proper noun.
no