Romans 9:17 says:

For the Scripture says to the Pharaoh, "For this very purpose I have raised you up, that I may show My power in you, and that My name may be declared in all the earth."

So I asked the question, could Pharaoh repent?

Ben did a great job with this and took the same track I would.

In Exodus 4, before Moses left for Egypt, God told him

"When you go back to Egypt, see that you do all those wonders before Pharaoh which I have put in your hand. But I will harden his heart, so that he will not let the people go. Then you shall say to Pharaoh, "Thus says the LORD: "Israel is My son, My firstborn. So I say to you, let My son go that he may serve Me. But if you refuse to let him go, indeed I will kill your son, your firstborn.""' (v. 21-23)

In Exodus 7, God instructed Moses again what to say before Pharaoh. In verses 1-5, the Lord says:

"See, I have made you as God to Pharaoh, and Aaron your brother shall be your prophet. You shall speak all that I command you. And Aaron your brother shall tell Pharaoh to send the children of Israel out of his land. And I will harden Pharaoh's heart, and multiply My signs and My wonders in the land of Egypt. But Pharaoh will not heed you, so that I may lay My hand on Egypt and bring My armies and My people, the children of Israel, out of the land of Egypt by great judgments. And the Egyptians shall know that I am the LORD, when I stretch out My hand on Egypt and bring out the children of Israel from among them."

This is a clear demonstration of Proverbs 21:1:

The king's heart is in the hand of the LORD, Like the rivers of water; He turns it wherever He wishes.

Exodus 8:15, 32; 9:34 do say that Pharaoh sinned and hardened his heart. But 9:12; 10:1, 20, 27; 11:10; 14:8 all say that the Lord hardened Pharaoh's heart. Notice the wording of Exodus 10:1-2:

Now the LORD said to Moses, "Go in to Pharaoh; for I have hardened his heart and the hearts of his servants, that I may show these signs of Mine before him, and that you may tell in the hearing of your son and your son's son the mighty things I have done in Egypt, and My signs which I have done among them, that you may know that I am the LORD."

The Lord hardened Pharaoh's heart that He might show His signs... that He may be known as the Lord. God's sovereignty over Pharoah's heart is evidence that God is God. What does that say of someone who would deny God's sovereignty over hearts?

Isaiah 6:10 says

"Make the heart of this people dull,
And their ears heavy,
And shut their eyes;
Lest they see with their eyes,
And hear with their ears,
And understand with their heart,
And return and be healed."

God warns Isaiah that his message will "fall on deaf ears." In fact, it would further harden their hearts and lead to their destruction. And these were God's chosen people, the Israelites, to whom he was going. Sometimes the preaching of the gospel makes the world worse. It is not the results that please God, but the obedience. It is God who will harden hearts and God who will open hearts.
And 63:17 says

O LORD, why have You made us stray from Your ways,
And hardened our heart from Your fear?
Return for Your servants' sake,
The tribes of Your inheritance.

They are not suggesting that they were innocent until God caused them to sin. They are acknowledging that they did indeed sin and are personally responsible for that sin. They are naturally sinful. "And even as they did not like to retain God in their knowledge, God gave them over to a debased mind, to do those things which are not fitting" (Romans 1:28). They are born sinful and God gives them over to more sin because of their sinfulness. But "He has mercy on whom He wills, and whom He wills He hardens" (Romans 9:18).
Jacob and Pharaoh were both born sinful. God had mercy on Jacob out of His own free, independent will. But God hardened Pharaoh to his own destruction. In both cases, God was given the glory and shown to be truly God.
JD brought up Judas. Judas is a great example of God's sovereignty and man's responsibility. Peter put it this way in Acts 2:22-24:

"Men of Israel, hear these words: Jesus of Nazareth, a Man attested by God to you by miracles, wonders, and signs which God did through Him in your midst, as you yourselves also know-- Him, being delivered by the determined purpose and foreknowledge of God, you have taken by lawless hands, have crucified, and put to death; whom God raised up, having loosed the pains of death, because it was not possible that He should be held by it."

What they did with lawless hands (personal responsibility) was done by the determined purpose and foreknowledge of God (divine sovereignty).

Now, for the next question. Paul asks it in Romans 9:19:

You will say to me then, "Why does He still find fault? For who has resisted His will?"