King of Egypt

How can the later king of Egypt not have known that Joseph was the one who saved his ancestor's reign?
During the four centuries that the Hebrews spent in Egypt, the country went through two political upheavals. In the 17th century BC, all but the southern end of Egypt was conquered by the Hyksos, a Semitic people who came from the (Middle) East. There is dispute on whether this conquest was military, enabled by superior weaponry (e.g., possible introduction of the chariot, sharper arrow tips and stronger bows made of composite materials) or a more gradual migration lasting decades (less likely), but there is no dispute over the fact that the Hyksos reigned for just over a century until 1570 BC, when Ahmose led the Egyptians in the south to push the Hyksos out of Egypt and established the 18th Egyptian dynasty with himself as the new Pharaoh. So the "another king" (Acts 7:18) mentioned above wasn't a descendant of the Pharaoh whose neck Joseph had saved, but was the descendant of Ahmose, "another" king.

Why did the later king deal "treacherously" (Acts 7:19) with the Hebrews in Egypt?
The Hebrew population had grown so fast that they outnumbered the Egyptians, and Pharaoh began to fear their uprising, especially if Egypt were to be attacked by another country: And Joseph died, all his brothers, and all that generation. But the children of Israel were fruitful and increased abundantly, multiplied and grew exceedingly mighty; and the land was filled with them. Now there arose a new king over Egypt, who did not know Joseph. And he said to his people, “Look, the people of the children of Israel are more and mightier than we; come, let us deal shrewdly with them, lest they multiply, and it happen, in the event of war, that they also join our enemies and fight against us, and so go up out of the land.” Therefore they set taskmasters over them to afflict them with their burdens. And they built for Pharaoh supply cities, Pithom and Raamses. But the more they afflicted them, the more they multiplied and grew. And they were in dread of the children of Israel. So the Egyptians made the children of Israel serve with rigor. And they made their lives bitter with hard bondage - in mortar, in brick, and in all manner of service in the field. All their service in which they made them serve was with rigor. (Exodus 1:6-14)

Do you see any parallel today to the above phrase, "the more they afflicted them, the more they multiplied and grew"?
This is what persecution does to true Christianity - it expands it, like trying to put out an oil fire by throwing water on it.

How did the king make the Hebrews "expose their babies, so that they might not live" (Acts 7:19)?
He tried to have the Hebrew midwives kill all newborn boys. When that didn't work, he commanded that the newborn boys be thrown into the river: Then the king of Egypt spoke to the Hebrew midwives, of whom the name of one was Shiphrah and the name of the other Puah; and he said, “When you do the duties of a midwife for the Hebrew women, and see them on the birthstools, if it is a son, then you shall kill him; but if it is a daughter, then she shall live.” But the midwives feared God, and did not do as the king of Egypt commanded them, but saved the male children alive. So the king of Egypt called for the midwives and said to them, “Why have you done this thing, and saved the male children alive?” And the midwives said to Pharaoh, “Because the Hebrew women are not like the Egyptian women; for they are lively and give birth before the midwives come to them.” Therefore God dealt well with the midwives, and the people multiplied and grew very mighty. And so it was, because the midwives feared God, that He provided households for them. So Pharaoh commanded all his people, saying, “Every son who is born you shall cast into the river, and every daughter you shall save alive.” (Exodus 1:15-22)

What lessons do the actions of the midwives hold for us today?
When the commands of those who have authority over us contradict the commands of God, obey God rather than men.