Is it permissible to repeat the same surah after Al-Fatihah in the second rak'ah? Is it permissible to consistently recite a specific surah after Al-Fatihah in all prayers and rak'ahs? And is the prayer valid with that?

It is permissible for the worshipper to recite the same surah after Al-Fatihah in both the first and second rak'ahs. This is indicated by what Abu Dawud narrated concerning the Prophet's (peace be upon him) recitation of Surat Al-Zalzalah in the two rak'ahs, and what Al-Bukhari and Muslim narrated from Aisha that the Prophet (peace be upon him) approved of a man who used to conclude his prayer with "Qul Huwa Allahu Ahad" (Say: He is Allah, [the] One). This is permissible according to the majority of scholars, even though it is contrary to the Sunnah and the Prophet's (peace be upon him) regular practice of varying his recitation. Sheikh Ibn Uthaymeen clarified that this action is permissible but not prescribed, meaning that the one who does it is not condemned, but it is not enjoined upon him to do it.