Does Chanukah primarily commemorate the miraculous victory over the Greeks, or the miracle of the oil?

The Gemara relates that the jar of oil that the Chashmonaim found, was still sealed. But what Halachic difference did this make? If a Nochri had touched the jar or carried it, it would in any case be Tamei!

What parallels do we find for this miracle, in which Hashem bestowed blessing on an item of small quantity, increasing it to meet or surpass our needs?

What exactly was the miracle that took place involving the lights of the Menorah?

Why would it take eight days to obtain Tahor oil?

What is another way to interpret the name "Chanukah"? What is "Chein," and who merits it? How did Hashem grant us "Chein" through this miracle?