Is he sinful who delays the Fajr prayer, thinking that the time has not expired, while the time has in fact expired by a few minutes?

There is no sin upon you as long as you did not intentionally delay the prayer beyond its prescribed time, for Allah Almighty says: "And there is no blame upon you for that in which you have erred, but [only for] what your hearts intended," and the Prophet, peace and blessings be upon him, said: "Indeed, Allah has pardoned for my Ummah their mistakes, their forgetfulness, and what they are coerced into." It is nevertheless necessary to take measures that help one wake up for prayer.