Is it permissible to keep supplications (Du'a), remembrances (Dhikr), Qur'anic verses (Ayat), or Prophetic narrations (Hadith) in a wallet, knowing that most people place them in their back pocket where they are susceptible to being sat upon?

It is permissible to carry the Qur'an or verses from it in a wallet under certain conditions, and there is no harm in carrying supplications and hadiths, even if the carrier is in a state of minor ritual impurity (muhdith). As for leaning on the Qur'an and books of knowledge or using them as a pillow, there is a difference of opinion: the Hanafis disliked it if there was no intention of disrespect, the Hanbalis forbade it if it contained the Qur'an, otherwise it was disliked, and the Shafi'is forbade it without a need.