There's a subtle shift in how the narrative refers to Pharaoh and the army part way through the account. We can see the first method in the first question:
Q: Does the Pharaoh actually leave with the army to chase the Israelites?
Exodus 14:7 So he made ready his chariot and took his army with him, (ESV)
The Pharaoh is spoken of directly. Prior to this in 14:5 and again in v10 Pharaoh is the primary identifier.
"The Egyptians" is the other term used in v9:
The Egyptians pursued them, all Pharaoh’s horses and chariots and his horsemen and his army, and overtook them encamped at the sea, by Pi-hahiroth, in front of Baal-zephon.
The account uses both "The Egyptians" and "Pharaoh" at this point. Is Pharaoh included in "Pharaoh's horses, chariots, horsemen and army"? It would seem reasonable to include Pharaoh (with his chariot).
Ex. 14:10 When Pharaoh drew near, the people of Israel lifted up their eyes, and behold, the Egyptians were marching after them, and they feared greatly. And the people of Israel cried out to the LORD.
We see that Israel looked up and saw Pharaoh and his forces marching toward them. So Pharaoh leaves with the army and is with them when they draw near to the Israelites.
Q: Does the Pharaoh enter the sea with the army?
A: Possibly, even probably, but not certain
So it is at this point that who is being referred to becomes unclear. Ex. 14:23-24,28
23 The Egyptians pursued and went in after them into the midst of the sea, all Pharaoh’s horses, his chariots, and his horsemen.
24 And in the morning watch the LORD in the pillar of fire and of cloud looked down on the Egyptian forces and threw the Egyptian forces into a panic,
28 The waters returned and covered the chariots and the horsemen; of all the host of Pharaoh that had followed them into the sea, not one of them remained.
No longer is Pharaoh referred to directly. He may be included in "the Egyptians" but it is never "Pharaoh and the Egyptians". Instead, it is the "host of Pharaoh" and the "Egyptian forces" and the "horses, chariots, and horsemen." These references may include Pharaoh, but we can't say for sure.
Q: Does the Pharaoh die in the sea?
Verse 28, quoted above, suggests that all the Egyptians died, which would have to include Pharaoh. Or does it? At this point "all the Egyptians" are all of the Egyptians in the sea. It does not include all the Egyptians everywhere.
We must conclude that it is not definite that Pharaoh did die. It IS the plainer reading of the text, but it is certainly not demanded by the text.
If the text demanded Pharaoh went into the sea then it would also demand he (now being a part of "all the Egyptians") also died since not one of them remained. But if we cannot say definitively that Pharaoh enters the sea personally we cannot say Pharaoh drowned definitively.
Q: So who did die?
A: At least the army of horses, chariots, and horsemen
Let's go back to what God claimed was going to happen, before the crossing, in Ex. 14:18.
Ex. 14:18 And the Egyptians shall know that I am the LORD, when I have gotten glory over Pharaoh, his chariots, and his horsemen.”
So God is either saying:
- The Egyptians will realize the LORD as God just before they die
- All the Egyptians back in Egypt will hear of this and know He is God.
For #2 to be true, someone would have to survive to tell the story so that all the Egyptians may know.
There is not enough textual evidence to say for certain the Pharaoh entered the sea and died with his army.
However, a plain reading of the chapter would be that a Pharaoh is included in "the Egyptians" and included in those who died.
This should be our basic understanding, but it cannot be claimed definitively and certainly not used as a factual requirement when trying to identify who the Pharaoh was. If other evidence suggest that Pharaoh lived, we may need to adjust that understanding. Historical evidence is often used to inform our interpretation of scripture for better or worse.