Kindly refer to the following link(s): https://www.youtube.com/watch?v=4NfYvMI00Hs

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If you are in the bathroom, you must say it without moving your lips and the same applies to the dua after it.

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It is permissible to wipe over cotton socks and thin ones as well.

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It seems permissible as you are not doing it while making wudu. But if you believe that making your neck wet before or after wudu is preferable then it is b’idah.

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