Can God remove distress from a servant, or must he continue to atone for his sin for the rest of his life? And are there other deeds that hasten the removal of distress?

Your repentance to Allah -the Almighty- is hoped to relieve you from worldly calamities and erase your sins, for "the one who repents from a sin is like one who has no sin." Allah's testing of a servant is not solely a punishment for sins; it can also be for elevating ranks and purification. It is not important for a Muslim to know whether they are being punished for a sin or not. Rather, they should know that calamities are a trial through which Allah atones for misdeeds and elevates ranks, which should motivate them to be patient and content.