All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
The first narration which you mentioned in the question “Stop doing everything during the Athaan” is a fabricated narration and it is not reported in any of the authentic books of Sunnah. Another narration with a similar meaning is reported “Whoever speaks during the Athaan, it is feared that he will lose his faith.”; however, the scholars stated that this narration is fabricated, therefore it is not permissible to spread it.
Whoever circulates this narration or sends it to others has indeed participated in lying about the Prophet .
The scholars stated that it is permissible to talk when the Athaan is being called, and they stated that the companions used to talk when the Athaan is being called.
As regards the supplication “O Allaah, help me to remember You, to thank You, and to worship You in the best of manners.”, then this is a supplication that is said after the obligatory prayers and it is proven as an authentic narration as we mentioned it in Fatwa 81736. Hence, it is permissible to circulate this narration and inform the people that this is one of the legislated mention of Allaah after the obligatory prayers. This is considered among teaching beneficial knowledge.
Nonetheless, it should be noted that the Muslims should not be driven by this phenomenon of sending claimed Prophetic narrations by mobile phone messaging and urging the people to spread it. Such narrations could be fabricated, or that these messages may include innovations and superstitions and speaking on behalf of Allaah and His Prophet without knowledge. Sending those messages and spreading them before ascertaining their authenticity is a very serious religious matter.
Allaah Knows best.