1. No.
2. You can actually do both.
3. The basic idea is if you leave the United States for less than six months at one go usually you are okay, you are not questioned too closely, but more than six months they question you very closely because of certain technical reasons.
At one year outside you lose your green card automatically unless you have something like a re-entry permit. If on the other hand, you are a U.S citizen there is no requirement that you have to stay in the United States. You can stay anywhere in the world, never have to come back again and you would still be a U.S citizen.
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