Is it permissible to pray the Sunnah (supererogatory) prayers with the intention of making up for missed fard (obligatory) prayers whose number is unknown, and will the prayer be accepted despite frequent absentmindedness and intrusive thoughts during it?

One must safeguard the prayer and not neglect it. The majority opinion is that missed prayers must be made up specifically, according to one's ability, provided it does not cause undue hardship or harm. If the number of missed prayers is unknown, then one should estimate and make up what ensures certainty or a strong presumption that one's duty has been discharged. Some scholars hold that making up the prayers is not obligatory, and that abundant supererogatory prayers along with repentance and seeking forgiveness suffice. However, the قول الجمهور (majority opinion) is the one that is issued as a fatwa. Lack of devotion (khushu') does not invalidate the prayer, and it does not need to be repeated for that reason.