Does the prayer become void if a worshiper stands for the third unit of prayer (rak'ah) and begins reciting Surat Al-Fatihah (by saying "Bismillah"), then returns to the first Tashahhud out of forgetfulness?

If you return to the first Tashahhud out of forgetfulness, your prayer is valid. However, if you do so knowingly and intentionally, your prayer is invalidated, and you must repeat it. Your ignorance of the prayer's invalidity does not harm you as long as you know that returning is forbidden. Your beginning with the Basmalah is considered a beginning of recitation, according to the view that the Basmalah is a verse of Al-Fatiha.