There is a common misconception that arises from the use of the phrase "free will" without proper definition. "Free" in the context of "free will" is usually used to mean "unfettered" as in "unfettered will", this clearly is a logical impossibility, since I can have the "will" to fly unaided, but I cannot fulfill that will. So, "free will" in an objective, non-theological sense, does not mean the ability to do everything I wish, instead it means something closer to "I am free to desire anything". In a theological sense, especially in the Biblical context it means something importantly different.
"Free will" in the Biblical context usually more precisely refers to the question of "
Are humans free to choose to do good vs evil?" or "
Is a person unhindered in choosing their eternal destination (heaven or hell)?" We can see that narrowing the question down immediately makes it easier to find an answer in the text of the Bible.
Before I address the question of God hardening Pharaoh's heart, let's see what Jesus had to say on the matter:
John 8:34 Jesus answered them, "Most assuredly, I say to you, whoever commits sin is a slave of sin."
The mark of a slave is that they are, by definition, not free. Committing sin shows us we are not spiritually free in any respect. Paul then clarifies this further:
Ephesians 2:1 And you He made alive, who were dead in trespasses and sins,
The mark of one who is dead is utter inability to do, think, feel, decide, or anything other than.. well, rot and stink. A dead person has no will, and a spiritually dead person has no spiritual will. So, we can see right away that we definitely do not have an unfettered will, nor an unhindered will. We are slaves to sin, and sin has domain over our will and therefore our actions. This is why we need God to be the initiator, and why Christ says that he leaves the 99 sheep that are safe to go find and rescue the one that is lost. (Notice that it is us who are lost, not Christ, and we are not the ones who go out to look for and find Christ, it is He who finds us.) But again, Paul sheds some clarity in this, by quoting from the Old Testament.
Romans 3:10 As it is written:
“There is none righteous, no, not one; 11 There is none who understands;
There is none who seeks after God. 12 They have all turned aside; They
have together become unprofitable; There is none who does good, no,
And Genesis has the following to say:
Genesis 6:5 Then the Lord saw that the wickedness of man was great in the earth, and that every intent of the thoughts of his heart was only evil continually.
And Jeremiah this:
Jeremiah 17:9 "The heart is deceitful above all things, And desperately wicked; Who can know it?"
So, before being saved and redeemed, our will is only continually desiring evil, all the time. And it is only the common grace of God that restrains evil men from being as evil as they could be, and doing the evil they wish to do. This explains why we haven't wiped ourselves off the face of the earth in our evil. So again, our will is not unhindered, it is enslaved to sin.
As to your question about Pharaoh, we now see that when the Bible tells us that God "hardened" the heart of Pharaoh, it wasn't that God made Pharaoh do an evil that Pharaoh did not want to do. The word "hardened" itself gives that away. Just like clay hardens into the position it had before being hardened, meaning it is no longer malleable as it was before, so too was Pharaoh's heart hardened. It was hardened into the evil "shape" it already had before its hardening. What this means theologically is that God withheld repentance from Pharaoh, which can only be understood if you understand that repentance is a gift from God, and that only God can grant that repentance.
Most discussions about free will usually by necessity address the issue of the Biblical commands to repent and live holy lives. Or, "Why does the Bible say we must repent, if we don't have free will?" Well, because the command to repent does not hinge upon our ability to obey it. We don't say our secular laws should be abolished because we find some people unable to obey them, so too with God's perfect law. He commands us to repent because we must repent. But, as we saw above, if repentance is a good thing, we cannot do it, and nobody does it.. except God first grant it.
Adam and Eve
But, within Biblical christianity I believe there to be little to absolutely no doubt that God did grant humanity free will. Adam and Eve had completely unhindered wills, completely free to live holy lives, obeying their Creator, or.. not. They entered existence as perfect sinless beings, Created in the image of God. But they chose to sin, sin entered creation, corrupting all of it, and as descendants of Adam and Eve we now inherit their corrupted nature:
Romans 5:12 Therefore, just as through one man sin entered the world, and death through sin, and thus death spread to all men, because all sinned— 13 (For until the law sin was in the world, but sin is not imputed when there is no law. [...])
Adam and Eve (specifically, Adam) sold humanity off in slavery to sin. And even if we reject this position, we must still accept Jesus' statement that sinning means we are slaves to sin. Slaves with wills, but not free ones.
It is important to clarify also, that though God is sovereign over all creation, including the will of man, it does not actually mean that He controls the will of man at all times. Also, being omniscient, knowing everything that will happen is not the same as causing everything to happen. Though, again, God is sovereign, and omniscient, I just wanted to clarify that these things are not the same.
In summary, we do have a will, but it is bounded by what is possible, and the common grace of God, who does not allow us to be as evil as we could be, and as slaves to sin, we definitely do not have free will. Even after redemption, our will is a result of the new nature He gives us, the new heart (which is a Biblical metaphor for "desires", ie, the fruit of our will) so in a sense, even after being freed from sin, we still do not have a truly "free" will, it is still bounded by what is possible and by the sovereignty of God.
Hope that answers your question.