Let's begin with some obvious things. First, God is, as you said, sovereign over all things. That means anything that ever happens at all is either a direct result of something God did (e.g., creation out of nothing), or indirectly through secondary means that God providentially governs (e.g., "cause and effect," laws of nature, human decisions). Second, everything God does is for the express purpose of displaying His glory. God's glory is the summation of all His attributes. While this might seem arrogant and egotistical from a human perspective, it's perfectly in line with God, who is the infinite and eternal one.
With that in mind, why did God allow sin? Unfortunately, the Bible doesn't give us a direct answer to that one. I can't point to a verse that says, "God allowed sin because..." But if God does everything for the purpose of displaying His glory, then I think it's safe to say that allowing sin to enter into the world serves the purpose of displaying His glory. That's what the Apostle Paul says in Romans:
Sin allows God to show mercy and grace to the elect and to show justice and wrath against the non-elect. Earlier in Romans 9, Paul had this to say regarding Pharaoh, "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'" (Romans 9:17 NKJV). To our fallen, human ears that sounds cruel and unfair. Why didn't Pharaoh get a chance to repent? Why didn't God give him a second chance? Well, if you know the story from Exodus, God gave Pharaoh ten chances to repent, and Pharaoh hardened his heart each time.
Romans 9 is a hard chapter for people to swallow. It's hard because it shows God in His complete sovereignty over salvation, and it renders us (fallen human beings) as helpless and hopeless outside of divine intervention. Despite all of that, we cannot hide behind God's sovereignty and say, "Well, there's nothing I can do, God is sovereign and I am not," because Romans 10 speaks of our responsibility. After saying God, "has mercy on whomever He wills and He hardens whomever He wills," Paul goes on to say, "Because they did not seek it by faith, but as it were, by the works of the law. For they stumbled at that stumbling stone... [But] if you confess with your mouth the Lord Jesus and believe in your heart that God has raised Him from the dead, you will be saved" (Romans 9:32; 10:9 NKJV). So how does divine sovereignty and human responsibility work together? That's another tricky question, but the thing to understand is we cannot say that human "free will" is somehow outside of God's sovereignty. If we do, then God is not sovereign. Our "free will" operates within the sphere of God's sovereignty is such a way that providentially our free choices accomplish God's sovereign will.
This leads me to your next question: Did Adam and Eve have "free will" before sin? Now, I've been putting "free will" in quotes for a reason. I think "free will" is greatly misunderstood by most people. Most people think "free will" means I can choose whatever I want and that my choices are not determined by anything. I feel that is greatly mistaken. I believe that our choices are determined by our desires. If I desire chocolate cake and chocolate cake is available, then I will choose to eat chocolate cake. However, if I desire chocolate cake but I also desire to lose ten pounds, I have a dilemma. Which desire is stronger? My desire to eat cake or my desire to lose weight? Whichever desire is strongest at the moment of decision is the desire I will choose. So our wills are determined, we never make choices in a vacuum. Thus, "free will" means my ability to choose according to my desires is not constrained or restricted by outside forces. To use a real life example (and not the trivial example of chocolate cake). The current societal pressures to curtail freedom of speech may constrain my desire to speak my mind on certain social topics. I may choose to speak out and face the consequences of my actions, or I can let societal pressure curtail my freedom to speak my mind. Thus, we do not have complete "free will" to exercise speech in certain venues or on certain occasions.
With that definition in mind, did Adam and Eve have "free will." Yes. They had the ability to choose according to their desires. Moreover, because this was pre-fall, they actually had the ability to choose to obey God or not to obey God. That's what makes the Fall so egregious. Adam had the innate ability to choose to obey God, but he gave into the temptation of the devil. His desire to be "like God" overcame his desire to obey God. As a result of this, mankind has lost the ability to choose to obey God. We have been born according to Adam's (now) corrupt human nature. Due to indwelling sin, we no longer have the ability to choose to obey God. Our will, our desires, are inherently anti-God. We still freely make choices, the problem is our desires are thoroughly sinful, and thus, our choices are always sinful, and we can never choose to follow or obey God, unless and until God makes us alive through the new birth (John 3:3ff). So instead of saying we have a "free will," it's actually more appropriate to say that the will of fallen mankind is in bondage. After our conversion (or regeneration), we are back (somewhat) to the position of pre-fall Adam, we are able to obey God or not obey God. The slight differences are: (1) the indwelling Holy Spirit, and (2) the remaining sinful flesh. Thus there is a battle within the Christian to walk according to the Spirit or according to the flesh.
As I said earlier, these are good (and hard) questions that Christians have wrangled with for centuries. It's very easy to get our minds twisted into knots trying to figure out the riddle of divine sovereignty and human responsibility. Here's the secret, we will never comprehensively reconcile these two truths. I can't point to a discrete point and say, "Here's where God's sovereignty ends and my responsibility begins." Like the Bible is written by man, but also written by the Holy Spirit, or like Jesus is 100% God and 100% man, we have to accept these biblical truths on faith. Every single time human beings have tried to reconcile these seeming paradoxes, they have always come down into one heretical error or another. The history of the church is littered with men who have created error by trying to resolve a biblical paradox (by the way, a paradox is NOT a contradiction). When it comes to divine sovereignty and human responsibility, I usually let God be God and I take care of what I'm called to do. My calling is to preach the gospel. I have not been given knowledge as to who the elect are, so I preach to everyone indiscriminately. But because I believe in God's sovereignty, I know my preaching will be effective. It will either lead to the conversion of the elect or to the hardening of the non-elect. Either way God is glorified.
I hope this helps.
~Pastor Carl.