What is the ruling on the prayer of someone who prolongs the recitation of Al-Fatiha and the Tashahhud due to obsessive whispers (waswas), and does not complete them until after the Imam rises from bowing, or does not complete the Tashahhud before the Imam's Tasleem? And what is the ruling on Satan threatening with illness anyone who does not yield to his whispers?

The cure for waswas (whispers/obsessive thoughts) is to ignore them and not pay them any attention. Giving in to them leads to great evil. It is permissible for a person afflicted with waswas to adopt the easier opinion among the sayings of the jurists.

Regarding the ruling on one whose Imam bows before he has completed Al-Fatiha: Some scholars hold the view that one should complete Al-Fatiha even if the Imam rises from bowing. Others hold the view that one should bow in adherence to the Imam, and that not completing Al-Fatiha does not cause harm. Sheikh Ibn Uthaymeen preferred the second opinion.

As for sending blessings upon the Prophet (peace be upon him) in prayer: The obligatory part of it is saying: "Allahumma salli ala Muhammad" (O Allah, send blessings upon Muhammad), and anything more is a Sunnah. Some scholars even hold the view that it is not obligatory.

There is no need to repeat prayers in which Al-Fatiha was not completed or in which an extra rak'ah was added due to ignorance. So, abandon these waswas and do not give in to them.