What is the ruling on not searching the trash for cardboard on which there is a drawing of a Mushaf or Surahs from the Qur'an, after knowing of their disposal therein?

Glorifying, respecting, and safeguarding the words of Allah the Almighty is a religious obligation. It is not appropriate to write Surat Al-Fatihah or any other verses on cardboard boxes or similar items that might be discarded, as this exposes them to disrespect.

As for what happened to you, there is no sin upon you; rather, you will be rewarded for your intention to preserve the verses you saw. You are not required to search through the garbage bin due to the burden and hardship involved, which Islamic law has negated. Sheikh Ibn 'Uthaymeen indicated that it is not obligatory for every person to search for papers on the ground unless they see a verse or a Hadith on them.