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.
In that sentence, "Arctic" is the proper adjective.
In general, you do not capitalize the first word after an ellipsis unless it is a proper noun or the start of a new sentence. An ellipsis indicates a pause or omission, so the sentence typically continues in lowercase. However, if the ellipsis is used to indicate the end of a sentence and the next word begins a new sentence, it should be capitalized. Always consider the context when deciding.
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.