Does Allah forgive a Muslim's sins if he dies on a Thursday or Friday, or on the nights preceding them, or on one of the days of Eid al-Fitr or Eid al-Adha, or on the Day of Arafah?

The statement mentioned in the question has no evidence to support its authenticity. It is worth noting that whoever dies on the night of Friday or during its day is saved from the torment of the grave, and this is confirmed by a Hasan (good) or Sahih (authentic) Hadith. As for one who dies while in a state of Ihram for Hajj or 'Umrah, he is washed with water and lotus leaves, kept away from anything perfumed, and his head is not covered, because his Ihram is not invalidated by death. It has been authentically reported that Abu Bakr al-Siddiq (may Allah be pleased with him) wished to die on Monday, seeking blessings from the Prophet's (peace and blessings of Allah be upon him) passing away on that day.