Is the Fajr prayer valid if the worshiper forgets to recite the Tashahhud completely, or recites only half of it, then stands for a third rak'ah out of forgetfulness, then remembers during it, says the Tasleem, and performs the prostration of forgetfulness (sujud al-sahw)?

If doubt arises after the salām, it should be disregarded. However, if it arises before the salām, then the prayer must be repeated if the worshiper has given the salām while doubting whether a pillar was performed and it was not, and a long time has passed, the prayer is invalidated and must be repeated.