Is what is stated in the message describing what the Quran does when a person dies — that a handsome man stands beside the deceased, enters between the shroud and the chest of the deceased, accompanies him when the two angels question him, and says, "I am the Quran that you used to recite" — authentic, or are these weak hadiths? And is the hadith mentioned therein, that the Quran will come on the Day of Resurrection as an intercessor for its companions, authentic?

We have not come across the wording mentioned in the question, but it has been established that the Qur’an is an intercessor whose intercession is accepted, and a truthful disputant. It will come as an intercessor for its companions on the Day of Judgment, preceded by Surah Al-Baqarah and Surah Al Imran. Also, reciting Surah Al-Mulk saves one from the torment of the grave. It has been reported that the Qur’an will be an adversary to whoever neglects its obligations and commits sins, leading them to be cast into the Fire. And it will be an intercessor for whoever adheres to its limits and acts in obedience to it, clothing them in robes of brocade and a crown of sovereignty, and quenching their thirst with the cup of sovereignty.