I'll start with your last question: Draco is considered semi-mythical, and we are not entirely sure whether he existed or not. However, most historians do believe he existed, or at least it is very probable, and he is credited with having codified Athens' first written law - probably in 621/620 B. C. E. Before, law was passed on by oral tradition, rituals, norms. It was sung, it was remembered by professional "rememberers", so-called mnemones. In the 7th century B. C. E., the first written law in Greece appear.

Draco himself was either called upon specifically to function as a law-giver, or he was already part of the thesmotéthai, (part of the árchotnes), the highest state officials in Athens, who replaced the monarchy.

As far as Draco's laws are concerned, we only have one law that was undeniably his doing - the law on homicide. This was Athens' first written law and as such had high symbolic value for the polis. This written law was written on blocks of wood which could be rotated and read - so-called axones - and exhibited on the agora, the central place of any Greek city/polis, where the people gathered.

Because it was made of wood, it did not endure. The original text did not survive, only a version that was written 200 years later! In 409/410 V. C. E. all existing laws in Athens were republished, including Draco's law. Thus, it is not clear how close to the original law this 409 version actually is. It might be completely similar, or some changes might have been made.

This particular law is on homicide, or rather it differentiates between murder and manslaughter or involuntary homicide. However, today's legal definition is not applicable to the understanding of Athens' people at the time! Essentially, this law was so important because Athens needed different punishments for actual murder and involuntary killing/death. Furthermore, this was connected to the tradition of blood feud/vengeance. The process of blood feud was problematic at the time, as it was not well regulated and led to feud upon feud between Athens' families. Draco tackled this by ruling quite in detail what punishment was to be given in which case, and he regulated and codified the process of blood feud.

Essentially, this is what we are certain of. Draco did write this law, and it played an important part in regulating Athenian society. However, scholars still debate, how many further law codes we are missing. Did Draco only codify this single law, or did he actually write a whole set of laws?

As I said, the problem is that the reproduction of this law and further commentaries were written centuries after the actual law. This led to a very one-sided portrayal of Draco's law(s) in the texts of Athenian authors. Most of them stated (much later!) that Draco's laws were so cruel that all of them except for the one on homicide had to be abolished, which was done by Solon, the second semi-mythical Athenian lawgiver who came after Draco.

Demades actually said that Draco's laws were written in blood (4th century B. C. E.)

Lysis said that Draco chose death as punishment even for the smallest offense (5th century B. C. E.)

Lycurgus also wrote that death was meant to be the punishment for all crimes in Draconian law (4th century B. C. E.)

As we can see, these portrayals were responsible for the view that Draco's law(s) were tyrranical and bloodthirsty. These portrayals were then copied and passed on from Antiquity to the Middle Ages. As a result, the phrase or term "Draconic/ Draconic law" is still in use today. However, we have no way of knowing whether is is true and just. Draco was also praised by others as a just lawgiver along with Solon. And as you see, the recipients in Antiquity already faced a complicated, fragmented source. Today's academic discussion reflects this and the debate on whether Draco only codified one law or multiple laws (possibly on theft and even education) is still ongoing.

So as often is the case, I cannot give a clear, satisfying answer, and neither can the experts on this topic. There actually aren't a ton of publications on this, but if you want to read what the experts have to say, the most important volumes are probably:
Gagarin, M.: Drakon and Early Athenian Homicide Law. London 1981.

Avengers of blood: homicide in Athenian law and custom from Draco to Demosthenes, Stuttgart 2008.

For contextual information on the emergence of written law from oral traditions, I would recommend articles by Rosalind Thomas and Karl-Joachim Hölkeskamp.