As the Ten Commandments were written in stone and writing had hardly developed anywhere else to that time, are those the first laws of civilization?
Nope. Even if we were to accept this definition of law as some written decree, and I'm unsure that's the case1, there are civilisations with written law that predate the Ten Commandments.
1. Most definitions of law don't require that it be written, but rather that it is some system of rules that govern the behaviour of some group of people.
Not with your age estimate of the Ten Commandments (3000-3750 years ago). The Code of Hammurabi dates to around 1750 BC, which would make it over 3750 years old, which is older.
Of course the first law are given to Adam, the same is true with Noah with a little modification and those are the same with the laws given to Moses. Shortly after The Great Flood, God directed Noah and his descendants to observe the following seven commandments, which are parallel or the same as that of the commandments given to Adam;
Do Not Worship a False Deity
Genesis 2:16 states: “And the Lord God commanded the (man) Adam, saying…” This Divine command to Adam implies that only the One True God, the Creator, should be obeyed and honored as the Deity, and the greatest honor is to serve and worship Him. Thus, one should serve and worship only the One True God, and not any idol.
Do Not Commit Blasphemy (Against your God).
Leviticus 24:10-17 relates the incident of a Jew who violated the injunction of Exodus 22:27 and blasphemed in anger. Moreover, it states in Leviticus 24:15, “ishish” (any man) who curses his God shall bear his sin.” Why the double expression of “ishish” (literally: “a man, a man”)? This includes all mankind, Jews and Gentiles. This demonstrates that blasphemy is thus prohibited to Gentiles even as it is for the Jews.
Do Not Commit Murder or Injury
The edict against murder, and the punishment for this transgression, is stated in Genesis 9:6: “Whoever sheds the blood of man, among man, his blood shall be shed; for in the image of God, He made man.”
Do Not Have Forbidden Sexual Relations
Five of the six types of relations that are forbidden by God to Gentiles are covered in Genesis 2:24: “Therefore a man shall leave his father and his mother and cling to his wife, and they shall become one flesh.” This verse explicitly forbids relations with one’s mother, with a woman who has ever been his father’s domestic partner or certified wife, with a woman who is currently a domestic partner or certified wife of another man, with another male, or with an animal. A Gentile is also forbidden to have relations with his maternal sister, who is learned from Gen. 20:13: “Moreover, she is indeed my sister, my father’s daughter, though not my mother’s daughter; and she became my wife.” (Note that Abraham said this to appease Abimelech. It was actually only figuratively true in his case, since Sarah was the daughter of Abraham’s brother. So they had the same paternal grandfather, who people often referred to as “father.”)
Do Not Commit Theft
The prohibition of theft is contained within the permission which God granted to Adam and Hava (Eve) in Genesis 2:16 to eat from the trees of the garden. This implies that if the permission had not been granted, they were forbidden to do, because the property did not belong to them. This applied specifically to the fruit of the Tree of “Knowledge of Good and Evil” which was forbidden for them to take, under penalty of death (Genesis 2:17). This Noahide commandment was cited explicitly by Abraham in Genesis 21:25.
Don’t Eat Meat that was Taken from a Live Animal
Adam and Hava (Eve) were not given permission to kill animals for food, and this remained in effect until after the Flood. God permitted the eating of meat for the first time to Noah and his family after they left the Ark, the reason why God at that time added the seventh commandment, which prohibited the eating of meat that was severed from a living animal (even if it was stunned and insensitive). This commandment given to Noah is recorded in Genesis 9:4: “But meat, with its soul [which is in] its blood you shall not eat.”
Establish Laws and Courts of Justice
God commanded Noah regarding the trial and punishment of a murderer, as it says in Genesis 9:6, “Whoever sheds the blood of man, among man; his blood shall be shed…” This refers to a Noahide commandment to judge and penalize a murderer. This is explained as follows by the Talmudic Sages: “Whoever sheds the blood of man” (referring to the murderer), “among man” (i.e., he is to be prosecuted in a court by a man who is qualified to testify as a witness), “his blood shall be shed” (if convicted, he is liable to capital punishment by the court).
The Noahide Code specifies that Gentile societies are obligated to abide by justice through establishing a system of righteous courts of law.
First, is the issue that, to the best of my knowledge, there's no actual evidence the ten commandments are real (in the form of stone tablets anyways -- obviously some form of them exists in literature). So it's hard to call "fair" about something that's quite possibly a fairy tale.
Second, even in the Bible story, it's not like God invented the concept of law then and there. The pharaoh obviously had his own laws, and the gods of those guys the Hebrews were partying with during the first writing of the commandments obviously had their versions of law.
Third, and most importantly, "law" isn't about written words, and never has been. Law is an abstract concept that represents the relationships between actions and consequences. The laws of physics existed long before we did, and law of the jungle existed between then and now.
The laws of society were an evolution from the right of might to what we have today, not some specific event. While it might be interesting to determine the first written law, the law itself existed long before then. Many societies have legal systems but no writing system at all.