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I may be wrong but,
If you drive any transistor with AC the out put wil match the input, you will get a modulated signal.
Yes, MOSFETS and IGBTs are off with no V applied. However their internal draw is so low you need to place a "pull down" resistor from gate to ground. For your application 100K should be low draw. Without this the applied 12V gate signal will take some time to slowly decay before it shuts the output fully off.
I dont think MOSFETS can handle reverse polarity, but if they can it would be cool. I havent worked with them before so I can't say. Testing with 5V will prevent most components from frying if something is wired wrong. Old computer power supplies have 5 and 12V leads and can be found in the trash for my favorite price $0.00 !
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