No there's no requirement for it to be capitalized.
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In general, "country music" is not capitalized unless it is part of a title or a specific name. For example, you would write "I enjoy country music," but "I attended the Country Music Awards." Therefore, "country" remains lowercase in standard usage.
Not generally, and it can look quite pretentious if you do.
It's a proper noun, so the first letter should be capitalized. Whether it should be written as "the Beatles" or "The Beatles" is hotly debated!
country music
Country music, pop, and rock