It is capitalized Western Europe
You should only capitalize it if at the start of a sentence or part of a title/name (Eg. United States Department of the Treasury).
If thename of the waltz is Wester n, the n it is capitalized. It should be Wester n Waltz.
When the w is capitalized.
'US' must be captialized, but for the 'the' you should only capitalize it if it is at the start of a sentence (Eg: The US is a big country).
You capitalize the "s" for state when you referring to a specific state or state agency; State of Texas or State of Maine or The State Dept. of Health. You don't capitalize the "s" when the word state is used in general terms; the states with the largest population or the western most state.
Yes, you should capitalize 'President' when it refers to a specific person ('The President of the United States'). It should be left uncapitalized when it is non-specific (Past presidents have included ... ').
Doesn't need to be, but you see it both ways, as well as Western hemisphere.
it depends on how you are using it. if it is part of a company's name then yes but if your talking a bout a region like southern Alabama then no.
Yes, you should capitalize the C in Century.
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.