What is the ruling on discarding papers containing the names of prophets, or the names of Allah, or Qur'anic verses and Prophetic hadiths into the wastebasket, and how should they be dealt with?

Respecting everything that contains the names of Allah Almighty and the names of the Prophets is obligatory by consensus. This includes everything glorified by the Shariah and all written material, especially that written in the Arabic language, as some scholars hold that letters possess an inherent sanctity. As for "Al-Awwal" (The First) in "Al-Sual Al-Awwal" (The First Question) and "Al-Salam" (Peace) in "Masirat Al-Salam" (The March of Peace), these are not among the Beautiful Names of Allah. It suffices to cross out Qur'anic verses with an ink pen when intending to dispose of them, and burying them in a safe place is more respectful.