1) It means that the mother reached the subject by circuitous verbal routes. In other words, she managed to make her point, and extract her promise, by speaking unclearly or deceptively on the matter at hand.
2) To be "survived" by someone is that a person lives after another dies. When my mother dies, she will be survived by me. When your parent dies, that parent will be survived by you and by the other members of your family.
Hope this helps!
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