Is the fate of a major sinner whose good deeds outweigh his bad deeds subject to divine will or to a balancing of good deeds against bad deeds, and does he enter Paradise directly or must he first be purified?

There is no problem in the fact that the perpetrator of a major sin is subject to the will of Allah Almighty. If Allah wills, He will forgive him out of mercy and grace; and if He wills, He will punish him for his sin out of wisdom and justice. The punishment may occur in this world, or in the grave, or in the plains of resurrection, or by the torment of the Fire followed by Paradise. Sheikh Al-Islam Ibn Taymiyyah mentioned that there are ten reasons for averting the punishment of the Fire from a believer: repentance, seeking forgiveness, effacing good deeds, the supplication of believers for him, their dedicating righteous deeds to him, the intercession of the Prophet (peace and blessings of Allah be upon him), expiating calamities in this world, the Barzakh (interim period between death and resurrection), and the Day of Judgment, and Allah’s forgiveness by His mercy. He also mentioned impediments to the fulfillment of the threat, such as repentance, seeking forgiveness, effacing good deeds, worldly afflictions, the intercession of the intercessor, and the mercy of the Most Merciful of those who show mercy. He said that the absolute threat is conditioned upon the absence of impediments and the presence of conditions. Al-Suyuti mentioned that believers are not punished for their sins except under specific conditions, such as the sins being major, dying before repentance, the absence of outweighing good deeds, and the absence of intercession for them. Ibn al-Qayyim said that Allah nullifies the effect of prohibitions through several matters, including: repentance, seeking forgiveness, effacing good deeds, expiating calamities, the angels' seeking forgiveness, the supplication of believers, the severity of death, the terror of what is to come, the severity of the standing (on the Day of Judgment), the intercession of intercessors, and Allah's mercy. If these matters are not sufficient, he enters the Fire until he is purified.