Shalom and thank you for your question! If I understood correctly, you want to understand what exactly is the Oral Law comprised of and from where do we learn the Halachot for our lives as practicing Jews? You were told that the Mishna is the Oral Law but you find this confusing because of the differing opinions. If there are just arguments- where are the laws?
Indeed the Mishna is the basis of the Oral Law, but to understand the answers to your quandaries we need to look at the development of the Oral Law from Mount Sinai onwards. We just recently read the Torah portion of Ki Tissah which contains the story of the sin of the golden calf and the subsequent breaking of the Tablets. G-d then tells Moshe Rabbeinu to carve the new Tablets himself. The Midrash explains that this is because he broke the first set, so G-d tells him to carve the second set. The second set of Tablets was given after G-d accepted the Teshuvah – the regret for their sin and the return of the Jewish people to G-d. This phenomenon of Teshuvah caused the people to merit a special gift with the second Tablets – Halachot, Midrash and Aggadot, which Moshe received and taught to the people. G-d told Moshe to write down the Five Books of Moses – Chamisha Chumshei Torah, but the rest was transmitted orally. The Chumash was followed by Neviim and Ketuvim, which were followed by the Mechilta, Sifra and Sifri, and the book of Zohar and the rest of the Midrash. The Mishnah was written down after that.
The discussions – sometimes heated – in the Mishnah are not generally on basic principles of Halacha, but matters of custom. The differing opinions are very important but I will address that soon. The Mishnah provides the basic structure of the Oral Law but the Halachot must be decided according to the principles laid out there, in the context of a particular case. That is why the Talmud discussed the Mishnah and later authorities like the Rambam and the Shulchan Aruch, codified the law based on those discussions. Each generation in fact adds to the body of Halacha because there are new issues all the time which need to be resolved – according to the principles laid out in the Mishnah.
There are ten soul powers. The first three are powers of the intellect, and the next seven are powers of the emotions. Every person is a combination of those soul powers, or energies. Some people could be powered mainly by the energy of Chessed for example. Chessed is translated often as loving-kindness. It is an energy of going towards people. Gevurah – called severity, is an energy of holding back. The house of Shamai in the Mishnah goes according to the energy of Gevurah, while the house of Hillel goes according to the energy of Chessed. Thus they often oppose each others ideas. During the long galut we need the energy of Chessed to reign supreme, but when Moshiach comes, laws will go according to the house of Shamai.
So different opinions have different purposes. There is a story about a Jew who was suffering from a lung ailment, and he went to his Rebbe for a blessing, since the doctors were very pessimistic about his chances for recovery. The tzaddik told him to go to the Holy Land to live, because according to the Talmud Yerushalmi, an animal with a similar ailment to that which he was suffering from, is considered kosher. Indeed this Jew went to live in Eretz Yisrael and recovered from his ailment. If I remember correctly he lived another twenty years. In another story a Rabbi was asked why he spent so long and consulted so many books in order to determine the status of one chicken, to see whether or not it was kosher. The Rabbi answered: If you could see how that chicken is pleading with me to pronounce him kosher you would understand!
The different opinions represent different spiritual levels. Different circumstances in life could lead to the need for Halachic decisions to be made in one way, and sometimes in a different way. However they are always based on the same basic principles. The same problem can a have a solution that is relevant in one context, and another solution that is relevant in a different context. In truth, we cannot really understand the Written Torah without the Oral Torah. We are not going to G-d forbid gouge out someone’s eye because it is written ‘an eye for an eye’. Rather, the Oral Torah explains that if we caused someone damage G-d forbid, accidentally or otherwise, we owe them restitution for medical and other expenses that they incurred as a result of that damage.
In Talmud Yerushalmi, tractate Peah 2:4 it says:”…even that which a diligent student expounds before his teacher was given to Moshe at Sinai.” This is because the diligent student is basing his interpretation on the principles which were given to Moshe by G-d at Sinai.
