The sentence that best adheres to AP Style and contains no punctuation, spelling, or grammatical errors is:
2) Henry Martin ‘Scoop’ Jackson was an American politician who served as a U.S. representative and U.S. senator from the state of Washington.
This sentence adheres to AP Style because it uses the correct form of the indefinite article 'an' before 'American', uses lowercase for 'representative' and 'senator' (AP Style does not capitalize these titles when they follow a name), and uses single quotation marks around 'Scoop', which is preferred in AP Style over double quotation marks.