The media are divided on this issue.

“Media” is a collective noun, and in British English, plural verbs are used with collective nouns (if the group is acting as individuals). In American English, singular verbs are typically used with collective nouns. That said, “the media are divided” is not grammatically incorrect. It’s just a difference in dialect. Edit: clarified how collective nouns are used in British English.

Yes, it's correct (but so is "is"). Was the exam in British English? BrE often treats collective nouns (in this context, "media" functions as a collective noun) as plural, e.g., "the group are ...," "the team are ..." In American English, collective nouns are almost always treated as singular.

I would use "is". When used in this way it's generally considered to be singular. The word media is plural or singular depending on context. The example isn't incorrect, it's just not what would most commonly be used when referring to mass media as a collective.

"Media" is the plural of medium, just like "data" is the plural of datum. Technically they should both be treated as plural, but in practice it sounds obnoxious.

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It should be "is"

Media is the plural of medium. So yes.