Is the one who kills himself—who is threatened with eternity in Hell in the Hadith—outside the fold of Islam, or is the Hadith to be understood as referring to one who deems that permissible?

The People of Sunnah (definition) and Jamāʿah believe that the disobedient among the monotheists are under the will of Allah; if He wills, He will punish them, and if He wills, He will forgive them. If He punishes them, their ultimate destination is to exit the Fire, due to the frequently recurring (mutawātir) hadiths that affirm the entry into Paradise of whoever dies as a monotheist, and the exit from the Fire of whoever has an atom's weight of faith in their heart. Al-Nawawi emphasizes that the madhhab (definition) of the People of Sunnah is that the monotheist will certainly enter Paradise, and that the converging legal evidences from the Qur'an, Sunnah, and the consensus of the Ummah support this principle. Any texts that contradict this must be interpreted to align with it.