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First, LG means you operate within the law, even if it appears to be immoral. You still have your lawful ethics.
No, that's not what it means. LG is lawful good, not just lawful, which means that an LG character will try her best to stay on the path of both good and law, but that good will win out if the two conflict and every attempt to follow both good and law fails. If law won out over good, the character would be LN. What differentiates LG and NG is that NG will just break the law, while LG will do her best to uphold it, then break it when she realizes that, in this circumstance, she cannot in good conscience uphold the law. An LG character is not bound to never, ever, ever violate the law. If she was, she's never be able to fight an LE character. For an example of an LG character going against the law, look at Captain America in Marvel Civil War. He's as LG as they come, but he rebels, and the reasoning he gives for rebelling is the exact reasoning I can see any LG character giving for going against the law.