What should the worshipper do if his lower garment falls off during the imam's prostration, so he catches the second prostration but misses the first? And what should the worshipper do if he forgets a pillar or an obligation?

If the izar (lower garment) falls during prayer and the awrah (private parts) are not exposed, then there is no issue. However, if its falling results in the exposure of a part of the awrah, and a prostration (sajdah) is missed, it can be made up for, and the worshiper can rejoin the Imam, and there is no harm in that. The Hanbalis and Malikis stipulate conditions for making it up, otherwise the incomplete rak'ah (unit of prayer) is invalidated. The "wajib (definition)" (obligatory) for the Malikis is synonymous with rukn (pillar), while for the Hanbalis, neglecting a wajib due to forgetfulness or ignorance is compensated for by the prostration of forgetfulness (sajdat al-sahw). The Imam bears the prostration of forgetfulness on behalf of the ma'mum (follower) if the latter forgets something that is not a rukn.