Is weeping from joy upon the mention of Paradise and good tidings for believers, or weeping from despair over the inability to abandon sin, or weeping from the fear of God—whose eye shall not enter Hellfire?

Crying out of the fear of Allah is one of the means of salvation from the Fire, and this includes crying out of fear of sins or being affected by Allah's promises and longing for Him. Likewise, achieving Tawhid (monotheism) and dying upon it, martyrdom in the cause of Allah, and maintaining the Fajr and Asr prayers are also means of safety from the Fire.