Are the remembrances after Fajr prayer considered part of the morning remembrances, and are the morning remembrances recited again if they were recited after Fajr? And are the remembrances after Asr prayer part of the evening remembrances, and are the Mu'awwidhat (Surahs Al-Falaq and An-Nas) and Al-Ikhlas recited twice more, so they are recited three times, as part of the evening remembrances?

There is an overlap between the remembrances (adhkar) of prayer and the morning and evening remembrances. Ayatul-Kursi and the Mu'awwidhat (Surahs Al-Ikhlas, Al-Falaq, and An-Nas) are prescribed after prayers. So, whoever recites them after the Fajr prayer, they suffice for both the prayer remembrances and the morning remembrances. It is advisable to recite Al-Ikhlas and the Mu'awwidhat three times. And whoever recites them after the Asr prayer, they suffice for both the prayer remembrances and the evening remembrances. It has been reported from the Prophet, peace and blessings be upon him: "Say: {Qul Huwa Allahu Ahad} and the two Mu'awwidhat (Surahs Al-Falaq and An-Nas) when you enter the evening and when you enter the morning, three times; they will suffice you against everything."