My specific question is What is the reasoning for Hillel and Shami's disputes. Why did Hillel want to light 1, 2, 3 etc, and Why did Shami want 8, 7 ,6 etc?
The Gemara offers two possible explanations for the Machlokes. Either Beis Shamai lights to represent the days yet to come, while Beis Hillel lights to represent the days already passed. Alternatively, the Gemara suggests that Beis Shammai has the lights correspond to the 70 cows brought for Korbanos on Sukkos which decrease each day (13,12,11 etc.), while Beis Hillel applies the rule of Ma'alin ba'Kodesh, that we must always establish Mitzvos in a way that we increase the Kedushah rather than decreasing it.