Is a believer's loss of sanity a punishment from God? And will his repentance be accepted? And will he abide in Hellfire eternally? And can he be cured by his supplication?

There is a contradiction in your question. One who has lost their mind cannot repent nor can prayers be offered for them, because the basis of accountability has been removed. It is not necessarily true that the loss of intellect is a punishment, for Allah possesses profound wisdom in His actions. When a Muslim loses their intellect, accountability is lifted from them, as per the Hadith (definition): "The Pen has been lifted from three... and from the insane until they recover." If a person dies having lost their intellect, they will be judged according to what preceded the loss of their intellect. If they were among the people of Islam and their good deeds outweighed their bad ones, they are among the people of bliss. If their bad deeds outweighed their good ones, they are subject to Allah's will. Perhaps Allah makes the tribulation an expiation for sins. As for one who became insane before reaching the age of accountability and died in that state, the preponderant opinion is that they will be tested on the Day of Resurrection with a fire into which they will be commanded to enter. If they were among the obedient, they will enter it; otherwise, they will say, "From the Fire I flee." Praying for the afflicted that Allah may restore their intellect is hoped to be answered.