1. What is the Halachic definition of maaleh gera?
2. If the supposition in Judaism is that sanctity passed from G-d to Moses and down to the Sages, how can one explain the disagreements between the Sages. Sanctity is unique.
We have written about the definition of maaleh gera in Pamphlet 3. See there.
To make things easier we will cite the explanation again:
Rashi: maalat gera — brings up and vomits the food from its guts and returns it to the mouth to chew and grind it fine (Leviticus 11:3).
Rashbam: maalat gera — brings back its food up the windpipe after eating it (Leviticus 11:3)
Ibn Ezra: maalat gera — –[a word] derived from garo [throat], and maalat is a verb (Leviticus 11:3).
The definition of the word gera as throat is learned from the Mishnah (Yoma 2:3) where there is a listing of the sacrificial lamb’s organs. It is written there “the chest and the neck” and Rashi there explains, “v’hagera — this is the neck, and in it is connected to the windpipe, the liver, the heart, and the chest.”
As to your question of how we can reconcile the belief that the Oral Torah was passed from Moses at Sinai down to our day with the fact that the Mishnah and the Talmud are full of debates and differing opinions of the Sages:
The answer is that this is a logical contradiction but not a religious one. Factually there is no doubt that the creation of the Oral Torah was a human work which took place over the course of the first centuries BCE by melding the various and sundry opinions of the different Sages together, yet the believer must believe that these things were given to Moses at Sinai.
You must recall that faith is not based on facts or any rational basis. It is based on an ideological, ideal, faith-based world view which can contain within itself rational and factual contradictions.
For more detail, see our words on the portion of Shoftim and of Bo.