Must one repeat the Eid prayer if one made a mistake in the takbirat and recited Al-Fatihah, then performed the prostration of forgetfulness out of ignorance of the ruling on the follower's forgetfulness? And what is the ruling on a small part of the bottom of the foot becoming uncovered during prayer and noticing it without covering it?

The cure for whispers (waswas) is to disregard them and pay them no attention. Your prayer is valid if the prostration of forgetfulness (sujood as-sahw) was performed before the salutation due to ignorance of the ruling. If the prostration was performed after the salutation, then your prayer is valid by even greater right. As for the exposure of the soles of the feet, it does not invalidate the prayer, and this is the most preponderant opinion.