Is the prayer of one who is unaware of the ruling of all the actions of prayer according to the Shafi'i school invalid, such as one who believes that a Sunnah is obligatory or vice versa?

The prayer of someone who cannot distinguish between a fard (obligatory act) and a sunnah (recommended act) is valid by scholarly consensus. So, if one believes an obligatory act to be a recommended one, or vice versa, it does not harm their prayer. The same applies to someone who is ignorant of the fard and sunnah elements in their ablution (wudu); their prayer is valid by consensus. Similarly, it is valid for someone who does not hold any particular belief or does not know the difference between a condition (shart) and a pillar (rukn), yet performs the prayer correctly; their prayer is valid, as their knowledge that all of these elements belong to the prayer suffices.