Q1) He didn't remember, neither did I. He didn't remember, nor did I. Both of these work perfectly, but 'nor' is a little more formal than 'neither'.
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innovatiQ2)
1) Are either of them interested in your project? No, neither of them is interested in your project.
2) Have either of you broken my coffee cups? No, neither of us have broken your cups.
3) Will either of these rooms serve youas a study? Yes, I think that either of these rooms will serve me as a study.
4) Are either of your sisters abroad? No,
—I won’t/will not accept bribes. (I) —Neither/Nor will I5. (‘Nor’ is less formal.)