Yes and No depends on the sentence
Proper nouns. "Would you rather live nearer the Tropic of Capricorn or the Arctic Circle?"
No, unless it is at the start of a sentence.
Yes Major League Baseball because it's a subject
No, you don't.
Eskimos live in the Arctic.
The word "Arctic" is capitalized.
when do you capatilize the title President
Polar bears and penguins live in arctic areas.
No, although you would capatilize Heaven and Hell
In that sentence, "Arctic" is the proper adjective.
As a noun: "I am going on a trip to the Arctic to do a scientific study." As a verb: "Chicago has been experiencing Arctic conditions recently."
If you are reffering to a name, first letter, movie, or book, then yes.