You have asked a very good question since most Jews, including most Orthodox ones do not fully understand the concept of the Oral law. Lucky for you, there is someone here who fully understands it after many years of study.
The first question is where did the concept of the Oral Law originate from and what is the lesson of the oral law with respect to Judaism?
The origins of the Oral Law in fact predate the exodus and the Bible. In the early days of man, G-d transmitted information through prophesy and that was handed down orally from mouth to mouth from father to son. For example, G-d transmitted the Seven laws of Noah to Noah and Noah transmitted it by mouth to whoever would listen and to his sons. In fact, our forefather Jacob studied information about G-d from Noah's sons Shem and Ever. Even before the Bible, righteous men such as Noah and his sons, plus Abraham, Yitzchak and Jacob would think about G-d through studying discover new things about him. Abraham compiled the first piece of the Oral Law specific for Judaism since G-d loved Abraham and gave him incredible prophesy, this prophesy combined with his incredible mind discovered much information about G-d and knew concepts mentioned in the Bible before they were given. Even though the forefathers did not know the entire Bible, the were aware of some information contained in it through their insight. For example, the Bible says his son Joshuah married his second son to the wife of his first son after the first son died childless to bring seed to the one that died. This is called Yibum and Moses states this commandment later on in the Bible. There is also a commandment that if the daughter of a priest is a harlet she must be burnt. Compare this to Joshuah bringing out Tamar to be burnt for being a harlot. According to tradition, much more information was known by Abraham but much of it wasn't stated in the Bible. Abraham passed this information through his sons, Yitzchak uncovered even more information and passed in down to Jacob and on and on. From what I just told you it is now clear what the benefit of having the Oral law is, that it emphasizes a strong bond between parent and child, since the child must accept the Oral tradition from their parent from generation to generation or it ceases to exist. The oral transmission from father to son is emphasized in the Bible as being the key to Judaism, that one must honor their parents and "ask your father and he will tell you, your elder and he will tell you". Similarly "and your son will say "what's this?", and you will say to your son "with a strong hand the L-rd took us out of bondage..." Without honoring your parents and accepting their oral tradition, Judaism will cease to exist.
The second question is what is the oral law and why do we need an oral law
The Oral Law has two parts.
1) The belief that Moses explained the written Torah Orally to Israel, and that this information was passed down from generation to generation. This is the first part of the Oral Law, what Moses passed down.
2) The second part is the beauty of the Oral Law, and what is not known to most Jews. Jews believe that the Torah is infinite, since it is divine and an infinite amount of information is contained in it. It is therefore impossible to uncover all information about the Torah. Even if Moses knew all this information, it would take hundreds of thousands of years to record everything which would be impossible. Therefore, what G-d told Moses to write down he wrote down. So how do we go about giving an incomplete Torah to Israel? Moses himself made a brief summary of the Oral law that he understood and told this to Israel the most necessary information that it should easily be remembered. From the Written Law and Moses Oral Law, it is possible to uncover more information, and to expand the Oral Law in future generations. In every generation, there are thousands people who uncover new concepts in the Torah and expand the Oral Law, some more than others. And when this information is added to the previous Oral Law, even more information can be derived from it, so the Oral Law is an exponential function that expands at an exponential rate and as time progresses more is uncovered. As more is uncovered, arguments and differences of opinion arise as well. Much of the Talmud contains these expanded discoveries, and tries to use logic to prove which opinion is correct. The Oral law did not end with the Talmud, it continues to progress even today and simple people like me and you can uncover new concepts in the Torah and expand the Oral Law.
1. If the law as it is in the Mishnah was intended to be oral, then how would it be permissible to be written?
2. Why does the Mishnah quote many different opinions if it all came from one source?
3. Shouldn't the truth of the oral law given to Moses only be in one opinion? Not many opinions?
4. Why doesn't the Mishnah speak solely in the name of Moshe?
1. Since the days of old, one of the purposes of the Oral Law was to maintain the necessary bond between parent and child so that the oral tradition is passed from father to son. If the law was written down, then this bond would not be necessary, and this father to son tradition which is the primary component of Judaism will be mocked and every idiot and nation would make up their own version of the Torah and not heed to the traditions of their fathers, as is happening today. Therefore, it is forbidden to write down the Oral Law and the Oral Law was sung and memorized by all Jews. Also, if the Oral Law is written, Jews would not respect new Torah concepts that are discovered later on and are told Orally, such as is happening today and Jews would not put effort into expanding the Oral Law. However, the Oral Law expanded to such an extent and the greatness of the generations was declining that the Great Rabbis saw that the Oral Law began to be forgotten. Rather than have this precious Torah be forgotten they violated this prohibition on writing down the Oral Law, and as Tzivi says they found a verse in Psalms that they interpuited as allowing this in this time of need.
2,3,4. You are exaggerating the arguments in the Mishna. Usually (90% of the time or more) the Mishnah states a concept that is agreed by all Rabbis, and is the direct Oral Law transmitted through Moses. Even though the Mishna does not say explicitly that Moses said this in every paragraph, that is to not act repetitive as it does say this in the first verses of Mishna Avot that Moses transmitted the Oral Law to Moses who transmitted it to Joshuah who transmitted it to the elders.... and goes down the entire line and then quotes things stated by the person in the last line. Then it says that that Rabbi's student accepted the tradition and he says the following etc. So whenever the Mishna says something that is not argued on, it is directly stated by Moses. Basically, most verses in the Mishna start off with a general concept not argued with, this is the Oral Law of Moses. Arguments arise from three sides. 1) As the Oral Law is transmitted generation to generation, sometimes information is forgotten or unintentionally misquoted and an argument develops as to what is the correct transmission. This is the rarest form of argument, since in the Mishna that I read so far, 90% of the time, crucial information is agreed upon and argument arises only in the fine detals. 2) As the Oral Law is expanded, Rabbis make up a entirely new conceptual case not thought of before and differ how to apply the Torah to this specific case. 3) As the Oral Law is expanded, several Rabbis come up with different conclusions based on the previous Oral Law and Written Law such as specifics for Mosaic concepts such as what is considered carrying on Sabbath and similar things of this nature. Most arguments in the Mishna are of this type, where everyone agrees on the general concepts and argue on the specifics. Another Chunk of the Mishna deals with Rabbinical decrees which Rabbis made later on to guard the Biblical Prohibitions from being violated, and some times they argue on the extend of these prohibitions so don't confuse these with arguments on the Oral Law.