Is there a responsibility upon someone who wrote their name as "Abd al-Qadir" on government applications that may be thrown in the trash?

There is no harm in writing your name on papers and in transactions that you need to submit to others. The sin falls upon the one who discards papers containing the name of Allah into the trash. It is permissible to write the Basmalah on cards and letters, but it is not permissible for the recipient to throw them into waste bins or place them in an inappropriate location. The sin falls upon the one who disrespects these papers, while the writer bears no sin.