■ Did the Pharaoh of the Exodus die when the Egyptian army was destroyed in the Red Sea?

Yes, he did, though the book of Exodus does not pointedly mention that fact. It says:
“The Egyptians took up the pursuit, and all the horses of Pharaoh, his war chariots and his cavalrymen began going in after [the Israelites], into the midst of the sea. . . . Finally Jehovah said to Moses: ‘Stretch your hand out over the sea, that the waters may come back over the Egyptians, their war chariots and their cavalrymen.’ Moses at once stretched his hand out . . . And the waters kept coming back. Finally they covered the war chariots and the cavalrymen belonging to all of Pharaoh’s military forces and who had gone into the sea after them. Not so much as one among them was let remain.”—Exodus 14:23-28.
This account mentions the cavalrymen and military forces, but does not specifically say whether Pharaoh died. Nor does the Israelites’ victory song, in which they said: “Pharaoh’s chariots and his military forces he has cast into the sea, and the choice of his warriors have been sunk in the Red Sea.”—Exodus 15:4.
However, Psalm 136:1-15 shows that Pharaoh did perish. There we read of the giving of thanks by the people ‘to the One striking down Egypt in their firstborn ones, and the One bringing Israel out of the midst of them by a strong hand and by an arm stretched out, to the One severing the Red Sea into parts, and who caused Israel to pass through the middle of it, and who shook off Pharaoh and his military force into the Red Sea.’
So the book of Psalms complements Exodus and indicates that the haughty Pharaoh, who oppressed the Israelites, died in the Red Sea.