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 is His slave and Messenger.
It is not prescribed to repeat the Faatihah (Chapter 1) in the same rakʻah (unit of prayer); however, if the person repeated it due to forgetfulness, then he should perform sujood as-sahw (prostration of forgetfulness). If he deliberately repeats it, then some scholars held that such an act is disliked and others held that it is prohibited; however, it does not invalidate the prayer.
The Hanbali scholar Al-Mardaawi wrote, "It is disliked to repeat the Faatihah in the same rakʻah; this is the scholarly view of our school of Fiqh and the opinion held by the majority of the Hanbalis; many of them held that it is the decisive view in this regard. Some maintained that such an act invalidates the prayer..." [Al-Insaaf]
An-Nawawi wrote, "If the person recited the Faatihah twice in the same rakʻah out of forgetfulness, then there is no harm in that; however, if he does so deliberately, then there are two scholarly views in this regard. The correct and reported view is that the prayer is not invalid as a result of that because it does not change the form of the prayer in any way..." [Al-Majmooʻ]
The Maaliki scholar As-Saawi wrote about the actions that do not invalidate the prayer, "... such as repeating the Faatihah in the same rakʻah; this does not invalidate the prayer. However, it is prohibited to repeat it deliberately. If he forgetfully repeats it, then he should perform sujood as-sahw... " [Bulghat As-Saalik Li-Aqrab Al-Masaalik]
As for your second question, it is not clear. How can we describe what we have not clearly heard as ending or beginning? In any case, the preponderant view is that it is prescribed to recite the reported supplication after the athaan for the one who heard it as understood from the words of the Prophet when he said:
"Whoever after listening to the athaan says, 'Allaahumma Rabba hathihi ad-daʻwati at-taammah, was-salati al-qaa'imah, atee Muhammadan al-waseelata wal-fadheelah, wabʻath-hu maqaman mahmoodan al-lathi waʻadtah,' (which means: O Allaah! Lord of this perfect call and of the prayer that is going to be established, give Muhammad the waseelah and the fadheelah, and resurrect him to the best and the highest place in Paradise that You promised him), then my intercession for him will be allowed on the Day of Resurrection." [Al-Bukhari and others]
He also said, "When you hear the mua'ththin, repeat what he says, then invoke a blessing on me, for everyone who invokes a blessing on me will receive ten blessings from Allaah; then implore Allaah to bless me with the waseelah, which is a refined rank in Paradise fitting for only one of the servants of Allaah, and I hope to be that one. If anyone asks that I be given the Waseelah, then he will be assured of my intercession." [At-Tirmithi and others - Al-Albaani graded it saheeh (sound)]
"When a person sees the mua'ththin and knows that he is raising the athaan yet he cannot hear it himself due to distance or deafness, then the preponderant view is that it is not prescribed for him to repeat the athaan because this is particularly related to hearing it. The hadeeth clearly states that the reported supplication is to be recited upon hearing the athaan. This view is also supported by the analogy with invoking mercy upon the one who sneezes; it is not prescribed for the Muslim to invoke the mercy of Allaah upon the one who sneezes if he does not hear him say 'alhamdulillaah' (praise be to Allaah) after sneezing..." [Al-Majmooʻ]
Allaah knows best.