ANSWER: What you have heard is absolutely wrong.  No problem in doing other activities during athan including praying. The best however is to reply to the athan and recite the prescribed dua after it but this is not mandatory. 

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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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