Excellent question. In order to answer thoroughly, first we ought to establish a couple of baseline facts:
We have a limited capacity to understand everything. Christian doctrines regularly go well beyond the ability of human reason to conceive, most famously with the doctrines of the Trinity and of the Incarnation. So, whenever we ask a question about God or our faith, we must recognize that some truths are simply beyond comprehension.
"Free will" is not a Biblical idea. (This doesn't mean that free will is false.) The Bible never uses the phrase "free will", so however we choose to define free will, we are necessarily bringing an extrabiblical concept to the Bible, and we shouldn't be surprised if it doesn't fit perfectly.
How the Bible talks about responsibility and freedom
The Bible does make it clear that human beings have the ability to choose between good and evil, most clearly in Deuteronomy 30:11-15. And it is equally clear the God holds humans accountable for the choices we make (Deut. 30:16-20 and too many other passages to enumerate them all). Therefore we must believe that we are responsible for the choices we make.
However, the Bible never uses words "freedom" or "free" to describe the human capacity to do good or evil. Instead, the Christ saw doing evil as the antithesis of real freedom:
Jesus answered them, "Truly, truly, I say to you, everyone who commits sin is a slave to sin. The slave does not remain in the house forever; the son remains forever. So if the Son sets you free, you will be free indeed." (John 8:34-36, ESV)
The apostles spoke similarly, contrasting slavery to sin with freedom in Romans 6-8, Galatians 5, or 1 Peter 2:16. They saw freedom from sin as equivalent to slavery to God. For this reason, I like to drop "free" from discussions of the human capacity to make choices, and simply speak of the will instead of free will. However, for the sake of the present discussion I shall use the phrase "free will", with the caveat that this usage of the word "free" is a different concept from the Biblical concept of freedom.
Defining Free Will
In order to ask whether free will exists, we must first define what we mean by it. This is a harder question than it appears at first. Let's crystalize the question to a single choice: You find yourself faced with a choice between A and B. Your selection between the two options is either:
Determined by your existing beliefs, desires, habits, and perception of the situation, OR
it is not determined by those things, but is made "on a whim" without reference to your already existing state of mind.
In case 1, the choice is predetermined. An observer who knows your state of mind and the situation can predict accurately what you will do in advance (and note that God does know your state of mind, Matt. 9:4 Matt. 12:25, Psalm 94:11, Psalm 139:2,23, etc.). In case 2, your choice is indistinguishable from purely random coin toss. In either a random event or in your choice between A and B, no-one can know the outcome in advance, nor can anyone tell what the reason for the choice was after the fact. Neither an automaton nor a flipping coin seem like they can properly be said to have free will.
I define free will like this: The ability to consciously make decisions that have a real effect on yourself and the world around you. Note that this definition is compatible with either choice-making scenario above. The difference between you and a robot or a flipping coin is that you have the conscious experience of making a choice, while neither a robot nor a coin does.
Under this definition, the Bible clearly teaches that human beings have free will. We make choices, and these choices have real consequences.
This definition is incompatible with fatalism, which is the idea that your decisions have no impact on your situation. The classic example of the fatalist paradox is this: If you are ill and fated to die of the illness, it will do you no good to call a doctor, since you will die either way. In the other hand, if you are fated to recover, it also doesn't matter whether you call a doctor, since you will get better either way. Therefore, the fatalist must say that your actions (calling a doctor or not) is irrelevant to your outcome. In fatalism, you do not have free will because your decisions do not have any effect on your situation.
Answering your particular questions
As I showed above, the Bible clearly teaches that human beings have free will (in some sense) and that we are responsible for our actions. It also clearly says that we cannot blame our sins on God in any way:
Let no one say when he is tempted, “I am being tempted by God,” for God cannot be tempted with evil, and he himself tempts no one. But each person is tempted when he is lured and enticed by his own desire. (James 1:13-14)
I will leave the question of predeterminism for a little later down the line. But first, let me address the dilemma of your future daughter telling a lie in mid-December of 2050. If your two inferences are correct (i.e. that this means the sin is unavoidable and that your daughter is therefore not morally accountable for it), then it can never be the case that God knows about about a human sin in advance. However, the Bible gives many specific examples of God predicting specific people's sins. The two most famous examples are that God predicted Pharaoh would refuse to let the Hebrew slaves go (Exodus 3:19) and Jesus predicted exactly when and how many times Peter would deny him (Matt. 26:34). This means that it is not categorically unjust for God to hold people accountable for sins that he knew they would commit in advance. We can be sure of that even if we don't understand exactly how it works logically.
And God recognizes that this is very hard to understand. The book of Habakkuk is relevant to this question, and is well worth the read regardless. As a brief summary, the prophet Habakkuk complains to the Lord that the nation of Judah is perverting justice and neglecting righteousness with impunity (Hab. 1:1-4). God's response?
Look among the nations, and see; wonder and be astounded.
For I am doing a work in your days that you would not believe if told.
(Hab. 1:5)
God goes on to describe how he is going to bring the Chaldeans, a nation far more wicked than Judah, to inflict punishment upon Judah. This is indeed something to wonder at, something astounding, and hard to believe! And God goes on to say that he will hold the Chaldeans accountable for their (future) cruelty to Judah in 2:6-20.
The fundamental paradox here is summed up in Genesis 50:20: "You meant evil against me, but God meant it for good." God makes use of evil to accomplish good ends. Similarly, the Apostle Paul writes:
"We know that for those who love God all things work together for good, for those who are called according to his purpose." (Romans 8:28)
This does not mean that all things are good in themselves. There is undeniably a great deal of evil in this world. But they all work together for good, a fact which is accomplished by God. You might say that evil is so impotent that it cannot actually effectuate evil, at least not in the end. Consider for example the crucifixion of Christ: This was the greatest evil that any man has ever done. Judas and Pilate and Caiaphas conspired to kill Jesus, who was not only the man who uniquely did not deserve punishment of any kind, but also was and is God in the flesh. And yet, God used their evil deed to accomplish the greatest good in history: The redemption of mankind.
There is, of course, a paradox here: How can God hold human beings accountable for evil actions that God both knows about in advance and uses to accomplish good purposes? "Why not do evil that good may come?--as some people slanderously charge us with saying. Their condemnation is just." (Romans 3:8) The apostle Paul addresses this question in Romans 3 not with a logical resolution, but rather with an appeal to God's righteousness (3:4,7). The first section of Romans, which wrestles with many questions similar to yours, concludes with these beautiful words:
Oh, the depth of the riches and wisdom and knowledge of God! How unsearchable are his judgments and how inscrutable his ways!
For who has known the mind of the Lord,
or who has been his counselor?
Or who has given a gift to him,
that he might be repaid?
For from him and through him and to him are all things. To him be glory forever. Amen.
~ Romans 11:33-36
In short, we do not conclude finally that humans are responsible for their sins. This is a premise, told to us expressly by God. We don't need to derive it logically. How is God's justice compatible with his foreknowledge? Oh, the depth of the riches and wisdom and knowledge of God! How unsearchable are his judgments and how inscrutable his ways!
Predeterminism
Within classical Christian thought, there are two basic streams of thought on this question: Calvinism and Arminianism. Neither view, in its usual form, is compatible with open theism. (Note: Properly speaking, these terms are only applicable to non-Lutheran Protestants. In answering the question of predeterminism, Lutherans are approximately Calvinist while Catholics and Orthodox are approximately Arminian.)
Calvinism essentially accepts predeterminism as a fact. It does not contradict the definition of free will I gave above, though, as your choices being predetermined does not undermine your conscious ability to make a choice, and the fact that that choice has a real effect on the world. The dilemma of how we can be held responsible for predetermined choices is a mystery (in the vein of the incarnation or the Trinity), but it may be partially explained in the framework of God's revealed will vs. his hidden will. His revealed will is his laws and instructions - the things he has told us to do and not to do. We are held accountable for whether we follow the revealed will of God. And this makes sense to do, because with our free will, when we choose evil, we did it because we wanted to. God's hidden will, on the other hand, is the predetermined sequence of events that God knows in advance.
The movie analogy that you bring up is a useful one here. (Though I might suggest perhaps it's better to imagine a novel, wherein the characters are not being played by actors.) The characters make real choices within the framework of the story, and they are held accountable for them within the framework of the story. I don't suppose, when reading a book, when the villain receives his comeuppance, you would think, "He didn't deserve that! After all, his actions were all cause by the author and not free." It would be far more absurd for the characters of the story to think that. If we are characters in God's story, then we exist on an ontologically lower "level", so to speak, than him. His choices and our choices don't collide because they happen at different levels of reality. Thus, within our experienced reality, we are totally responsible for our actions, even though from a God's eye view, those actions are not only predetermined but willed by Him. (This isn't a perfect analogy, obviously, but it's useful.)
Arminianism rejects predeterminism, and says that human beings make real choices that are not willed by God, so there is no mystery about why we are held accountable for them. The fact that he knows them in advance and uses them to accomplish his will without predetermining them is a mystery. It is helpful to recognize that God exists outside of time, so he can see the whole timeline of the world and of your life at once.
A useful analogy to understand Arminianism (which I owe to C.S. Lewis, see Miracles Appendix B):
"Suppose I find a piece of paper on which a black wavy line is already drawn, I can now sit down and draw other lines (say in red) so shaped as to combine with the black line into a pattern. Let us now suppose that the original black line is conscious. But it is not conscious along the whole length at once--only on each point on that length in turn."
In this analogy, the black line is the human will, and the red lines are God's creation, which have been built up in response to the human will in such a way that they align with it and accommodate it. Thus the human choices are all integrated into God's plan, and known "in advance", while still being totally free and not caused by God in any way. (This analogy, too, is flawed.)
Recommended Reading
Within the Bible: I highly recommend you read Romans again. Try to read it in one sitting, to get the full sense of what Paul is saying. Then go back through more slowly and pay attention to the details of how he deals with this topic. Read Habakkuk as well, and Isaiah 45. Ephesians is also good for this topic.
Other books:
The Bondage of the Will by Martin Luther. I can recommend the translation by J.I. Packer and O.R. Johnson, their introduction is also quite insightful. This book is pretty dense, but it is well worth the effort of reading. One of my all-time favorites.
"Anxiety is the Dizziness of Freedom" in Exhalation by Ted Chiang. Not a Christian author, but I think he has some good insight into the problem of free will. His "story notes" in the back of the book are also worth the read. Let me quote: "In discussions about free will, a lot of people say that for an action of yours to be freely chosen--for you to bear moral responsibility for that action--you must have had the ability to do something else under exactly the same circumstances...When Martin Luther defended his actions to the church in 1521, he reportedly said, 'here I stand, I can do no other,' i.e., he couldn't have done anything else. But does that mean we shouldn't give Luther credit for his actions? Surely we don't think he would be worthier of praise if he had said, 'I could have gone either way.'" (p.349)
The Silver Chair by C.S. Lewis.
Miracles by C.S. Lewis.
The Hobbit by J.R.R. Tolkien. "Surely you don’t disbelieve the prophecies, because you had a hand in bringing them about yourself? You don’t really suppose, do you, that all your adventures and escapes were managed by mere luck, just for your sole benefit? You are a very fine person, Mr. Baggins, and I am very fond of you; but you are only quite a little fellow in a wide world after all!"
On the Predestination of the Saints by Augustine. I haven't read the whole thing myself yet, but from what I have read I can heartily recommend it for a greater understanding. "What do you have that you did not receive?"
