Is it obligatory to repeat the prayer or ablution (wudu') when the Imam makes mistakes or when touching the male organ during ghusl, and what is the effect of that on congregational prayer?

The default state for prayer is validity, and it is only invalidated by certainty. If a worshiper doubts the occurrence of something that invalidates their prayer, they should disregard this doubt and turn away from such whispers (waswās).

If the Imam commits a melodic error (laḥn) in Al-Fatiha that changes its meaning, the prayer is invalidated, and it is not permissible to follow him. However, if the error does not change the meaning, the prayer is not invalidated. If the error is in a part of the prayer other than Al-Fatiha, neither the Imam's prayer nor the follower's prayer is invalidated, unless the error changes the meaning and was intentional.

Whoever touches their private part during ghusl (definition) (ritual bath) has their ablution (wudu') invalidated, and they must perform wudu'. If they pray without wudu', their prayer is invalid. However, wudu' is only invalidated upon certainty of touching the private part. If it is merely a doubt, the wudu' remains valid.

A melodic error (laḥn) by the Imam in the opening takbīr (takbīrat al-iḥrām) does not invalidate the prayer.

The discomfort a worshiper feels during congregational prayer is from Satan to deprive them of its reward. Therefore, they should seek refuge with Allah and continue to attend congregational prayers in the mosque regularly.