Is it considered a sin not to inspect trash bags daily for papers containing sacred names, for fear of embarrassment or hardship? And is feeling shy in front of people when doing so an indication of sin?

It is not legislated to search through trash for papers containing the names of Allah, as this is blameworthy extremism. It suffices to advise family members of the impermissibility of disgracing such papers and the obligation to safeguard them, and to remove them if possible without hardship. As for the names of prophets, they must be respected if the intention is the prophet himself. The names of the Companions are not among those things that are divinely exalted in Islamic law.