Is it obligatory for someone who frequently experiences doubt in their prayer, ablution, and the number of their rak'ahs, and who feels that their fasting is invalid, to repeat all of their missed prayers and fasting?

You must turn away from misgivings and doubts and not pay attention to them. Do not repeat any dhikr unless you are certain you omitted it, and do not perform the prostration of forgetfulness (sujud al-sahw) unless there is a reason for it. Strive for humility (khushu') by pondering over the verses and dhikr. The prayer is not to be repeated due to lack of khushu'. You can refer to "Al-Mulakhkhas al-Fiqhi" by Sheikh Al-Fawzan, "Fiqh As-Sunnah," and the treatise "Sujud Al-Sahw" by Ibn Uthaymeen, as well as the fatwas on this website regarding the rules of prayer.