Funeral Prayer for one who committed suicide

Q: Is a Funeral Prayer to be offered on someone who intentionally killed himself with a pistol even though he had good posts such as a teacher or police officer and no one knows the reasons behind the suicide? If the answer is the affirmative, how can this be justified despite the authentically reported Hadith stating that whoever kills himself shall abide forever in Hellfire? Moreover, the Messenger (peace be upon him) said regarding Qizman who fought with the Muslims in the battle of Uhud and killed eight of the Kafirs (disbelievers): "He is in Hellfire"; because Qizman was wounded and when his injury became extremely painful, he took an arrow from his own quiver and killed himself therewith. Please mention the proofs that support your answer.Besides, is the Funeral Prayer to be offered on someone who made Zina (sexual intercourse outside marriage) with his paternal uncle's wife, killed his uncle with a knife, carried him at night, put him in a distant place, and burnt him along with the wife but was then arrested and killed as Had (prescribed penalty) was applied against him? Is a Funeral Prayer to be offered on such a person despite the saying of Allah (Exalted be He): And whoever kills a believer intentionally, his recompense is Hell to abide therein; and the Wrath and the Curse of Allâh are upon him, and a great punishment is prepared for him. Please clarify this issue in the light of the proofs that your Eminence has. I am aware that the Messenger of Allah (peace be upon him) offered the Funeral Prayer for the lady that committed Zina and requested that Had be applied against her and that her confession and desire to be penalized by being stoned to death was regarded as Tawbah (repentance to Allah) for the Messenger (peace be upon him) said: She has made such a repentance that if it were to be divided over seventy from the people of Madinah, it would have sufficed them. On the contrary, those who killed themselves did not make any Tawbah. Similarly, murderers are keen to hide their crimes and it is only Allah (Exalted be He) who helps the authorities to arrest them. May Allah help you to say the truth and peace be upon you.