Shortcut: semicolon + and/but = wrong
If you see an answer choice on either the SAT or the ACT with that construction, cross it out and move on. Why? Because a semicolon is grammatically identical to a period, and you shouldn’t start a sentence with “and” or “but.”
The slightly longer explanation: In real life, semicolon use is a little more flexible, and the choice to use when can sometimes be more a matter of clarity/style than one of grammar. It is generally considered acceptable to place a semicolon before “and” or “but” in order to break up a very long sentence, especially when there are already multiple commas/clauses.
For example: Pamela Meyer, a certified fraud examiner, author, and entrepreneur, became interested in the science of deception at at business school workshop during which a professor detailed his findings on behaviors associated with lying; and she subsequently worked with a team of researchers to survey and analyze existing research on deception from academics, experts, law enforcement, the military, espionage and psychology.
In this case, either a comma or a semicolon could be used before “and;” however, the sentence is so long and contains so many different parts that the semicolon is a logical choice to create stronger break between the parts.
Why not just use a period? Well, because a semicolon implies a stronger connection between the clauses than a period would; it keeps the sentence going. Again, this is a matter of style, not grammar.
The SAT and the ACT, however, are not interested in theses nuances. Rather, their goal is to test the quick and dirty version of the rule, to check whether you understand the most common version of the rule. Anything beyond that would simply be too ambiguous.