ANSWER: No, you must not make up that rakah.  You are exempted from reciting fateha and this rakah is valid if you caught the imam in ruku and you got to say the prescribed dhikr at least once. 

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ANSWER: You can recite it silently, providing you at least move your tongue and lips.

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ANSWER: No, you must not recite tashahhud after making sujood as sahu. You directly make salam.  https://www.assimalhakeem.net/sujood-as-sahu-prostrations-of-forgetfulness/

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ANSWER: There is no such thing as new year resolutions in Islam.  If you make an oath by taking Allah’s name then you must fulfill it. 

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ANSWER: If the water has reached in between your toes, pouring water is sufficient but the sunnah is to insert your fingers between your toes to ensure that. 

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