Praise be to Allah, and peace and blessings be upon the Messenger of Allah. To proceed:
The scholars have mentioned that the major events of the Day of Judgment occur approximately in the following order:
Resurrection and emergence from the graves, followed by people's movement to the land of gathering. They will remain standing there for a long and extremely difficult period, during which fear and thirst will become intense as they await the reckoning.
The Pool (Al-Hawd). The believers who remained steadfast upon the Sunnah of the Prophet ﷺ will come to it and drink from it. This will be among the first causes of security for them.
The Great Intercession. The Prophet ﷺ will intercede with Allah تعالى so that judgment between the creatures may begin and the reckoning may be hastened, after the people's standing at the gathering place has become prolonged.
The presentation and reckoning. The deeds of the servants will be presented, and they will be held accountable for them.
The scattering of the records and the reading of the books. The people of the right will receive their books in their right hands, while the people of the left will receive their books in their left hands. The proof will be established against the servants through what has been recorded in their records.
The Scale (Al-Mizan). The deeds of the servants will be weighed, thereby showing the excellence of the obedient and the loss of those who disobeyed.
Darkness and light. The believers will be given a light by which they will walk, while the hypocrites will be deprived of it and will remain with the disbelievers.
The Bridge (As-Sirat). People will pass over it according to their deeds. Some will cross quickly, some slowly, and some will fall into the Fire. The believers will cross according to the degree of their faith and deeds. Among the people of monotheism are those who will be punished in the Fire according to their sins and then emerge from it by the mercy of Allah.
The Bridge of Retribution (Al-Qantara) and retaliation among the believers. When the believers are saved from the Fire, they will stop at a place between Paradise and Hell, where some will settle their claims against others. Resentment will be removed from their hearts until they enter Paradise with pure hearts.
Entering Paradise or Hell. The people of faith will enter Paradise by the mercy of Allah. The disbelievers will enter Hell. Those sinful monotheists whom Allah wills will remain subject to His will: some will be forgiven by Allah, while others will be punished and then brought out by His mercy.
Therefore, the order mentioned in the question requires some adjustment. Most notably, the Pool comes before the Great Intercession and the reckoning, the Scale comes after the reckoning and the reading of the records, then the Bridge, followed by the settling of claims among the believers before entering Paradise.
We do not know of any authentic hadith establishing that two devils sit beside the dying person in the form of his father and mother and then command him to follow Judaism or Christianity. What has been transmitted concerning this from some books cannot be used as evidence, and therefore it is not permissible to assert that it is a hadith of the Prophet ﷺ.
However, it is established that Satan may attempt to mislead a person at the time of his death. The Prophet ﷺ sought refuge in Allah from Satan confusing or overpowering him at the time of death. This includes the possibility that Satan may exploit the severity of the death agony to turn a person away from repentance, cause him to despair, make him doubt his religion, or spoil his final state.
Some scholars have related accounts of righteous people who experienced satanic whispers at the time of death, but such stories cannot be used as evidence to establish the specific details mentioned in the question.
Therefore, the duty is to remain steadfast upon monotheism, seek refuge in Allah from Satan, think well of Allah, and prepare to meet Allah through obedience and repentance.
And Allah تعالى knows best.