What is the ruling on someone who stops performing an obligatory prayer without concluding it with the Taslim, then wishes to resume it? And what is the ruling on forgetting a prostration, a Surah, or anything else besides the Rak'ahs in an obligatory or supererogatory prayer?

The first taslima (salutation) is a pillar of prayer according to the majority of scholars, and it is a wajib (obligatory act) according to the Hanafis. Forgetting it invalidates the prayer according to the majority, unless it is rectified.

Omitting a prostration (sajda) from a fard (obligatory) prayer necessitates its rectification and the performance of the prostration of forgetfulness (sujud as-sahw). If rectification is not possible, the prayer becomes invalid and must be repeated.

The rulings regarding forgetfulness in supererogatory (nafila) prayers are the same as in obligatory prayers, but supererogatory prayers do not require qada' (making up).

As for forgetting a surah (chapter of the Qur'an), it does not invalidate the prayer according to the majority of scholars.