A few points:
-I wouldn't start by considering the sequence of events if I had other options. Verb tense is a tricky subject for many people, because verb tense rules have exceptions, and their exceptions have exceptions.
-The word "and" here is a parallel marker, "being a United States citizen since 1984
and born in Austria in 1947." That's promising but it may not get us too far. After that parallel marker we find the participial phrase "born in Austria." Participles can be parallel only with other participles (though past participles can be parallel with present), so "born in Austria" must be parallel with "being a United States citizen." Awful though that sounds, it's hard to say what is wrong with that grammatically, since the parallel elements are the same parts of speech. (To be clear, there are problems with this parallelism, but they're too subtle to struggle with during the test.) Every answer that preserves the "and" uses it to join two participles, and the answers that get rid of the "and" are not wrong for that reason, so parallelism may be a bust for most test-takers.
-So we end up where we usually end up, with splits. How do the answer choices differ? "Being born" vs. "having been born" vs. "born". Well, this brings us to TS's question about "being." "Being" is highly suspect; it almost always signals a wrong answer. Further, both "being" and "having been born" here suggest that Schwarzenegger was elected because he was born in Austria in 1947, which is surely wrong. That leaves only C. This is more or less a question of meaning/clarity, not my favorite place to start, but it'll do here.
-"Became" is the only verb in the first independent clause in C; the word "born" is not a verb but a modifier, so there's no verb tense issue in that clause considered alone. The sequence of events in the second independent clause--the one after the semicolon--would be clearer if the author used, say, a past perfect with a simple past. But we have to work with the answers given.
-The big takeaway here: Don't worry about a grammatical issue unless you are confident that A contains a specific error, or the grammatical issue is suggested by the splits. Don't worry about what the ideal sentence would be; it's almost never an option.