Can one who committed suicide, yet was a believer, enter Paradise, and what is the extent of the authenticity of the Hadith which mentions that the Companion who fell ill and severed the veins in his hand will enter Paradise, but Allah will not mend what he corrupted?

Killing oneself is a major sin, but its perpetrator is not deprived of entry into Paradise. The doctrine of the Ahl al-Sunnah is that a monotheist will not abide eternally in Hellfire. The eternal abode in Hellfire for one who kills himself refers to someone who considered killing himself permissible, thus becoming a disbeliever, or it is a deterrent, or it is a punishment that ceases when Allah extracts the monotheists from Hellfire. This is supported by the hadith of the man who migrated with Tufayl ibn Amr to Medina, then fell ill and became distressed, so he cut his finger joints and died. Tufayl saw him in a dream in a good state except for his hands. He informed the Prophet of this, and the Prophet supplicated for him: "O Allah, forgive his hands too." This indicates that the perpetrator of a major sin, who is not a disbeliever, will not abide eternally in Hellfire and is subject to Allah's will.