What is the ruling on the prayer of someone who intended to pray Fajr as a make-up (qada'), then changed their intention to the Sunnah of Fajr, then prayed it as a make-up, then led their family in it as a fard (obligatory) prayer, given the recurrence of these whispers (waswas) and doubts in every prayer? Do they (the family members) have to make up their prayer? And how can one get rid of these whispers?

As for the prayer you began with the intention of performing Fajr as a qada' (makeup) prayer, then changed your intention to Raghibat al-Fajr (the two rak'ahs before Fajr), it is invalid. The Fajr prayer you led as an Imam with the intention of performing the fard (obligatory) prayer, despite the whispers and doubts you experienced during it, is valid. You are not required to repeat it or inform the worshippers. You should not lead people in prayer until Allah removes the waswasa (whispering doubts) from you, even though the prayer of those who prayed behind you is valid. Morning and evening adhkar (remembrances) are a strong fortress for the Muslim, and abandoning them due to lack of contemplation opens the door to waswasa. You must constantly seek refuge in Allah Almighty to repel the waswas.