What is the ruling on someone who ate, thinking that the Fajr (dawn) prayer had not yet been called, and then it became clear to him while he was eating that the Fajr prayer had already been called? And does he complete his fast or not?

Your fasting in this case is not valid. You must make up for this day, while refraining from eating for the rest of the day, because the eating occurred during daylight hours. There is no expiation required of you according to the majority of scholars.