Is it true that the Mu'awwidhat (Surahs Al-Falaq and An-Nas) should be recited once after every prayer, and three times after the Fajr and Maghrib prayers, or are they recited only once after all prayers, and are they repeated three times only if they are part of the morning and evening remembrances (adhkar)?

Reciting the Mu'awwidhat (Surahs Al-Falaq and An-Nas) after every prayer is mentioned in an authentic Hadith narrated by An-Nasa'i from 'Uqbah ibn 'Amir. As for repeating them three times, it is mentioned in an authentic Hadith narrated by Abu Dawud and At-Tirmidhi from Abdullah ibn Khubayb, regarding reciting them in the evening and in the morning, where three times suffice for everything.

Therefore, reciting them once after every prayer is prescribed. Repeating them three times is only in the morning and evening, not after the prayers.