When the lamb was slaughtered, was this done at twilight, at the beginning of the founteenth, or at the going down of the sun at the end of the fourteenth? Can you please explain the time of the events?

The phrase in the Torah for the time of slaughtering the lamb is "Bein Ha'arbayim" which refers to the period between noontime, when the sun begins to decline in the sky, and sundown. The actual time was mid to late afternoon (see Mishna Pesachim ch. 5). The offering was eaten at a festive meal at night.