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: 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: No, you should only say Rabbana wa lakal hamd while rising up from Ruku. 

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ANSWER: If he missed the Friday prayer for a valid reason, he must pray 4 rakahs of dhuhr. 

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ANSWER: No, this is not permissible. If you pray it even one second before time, this would be invalid.  You must pray the Sunnah after the time of the fard prayer enters. 

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