Yoma 67a - dividing the piece of red wool

"He would divide the strip of red wool; half he would tied on a rock, and half between the two horns of the se'ir hamishtaleach..."
Even though the gemara there says that the man was allowed to carry the se'ir on Shabbos if need be, I don't understand why he would be allowed to divide the strip of wool. The first is an essential part of his job and therefore the Torah says that he can carry out his job even on Shabbos. But the strip of wool isn't mentioned in the Torah, it just seems to be learned out from a hint in Navi. And you could do it without splitting it, it's just that the gemara says that then there's a risk of not seeing whether it turned white.
It would seem to me that this is a real melachah, not just a d'rabanon. The man is מחתך, "cutting to shape", making two pieces of the shape that he needs.

I saw a pictorial guide to the Yom Kippur avodah, and it seemed from the picture that the "lashon" (tongue, strip) of red wool was a packet of fibers [probably washed, combed and dyed, but that's all], not a woven piece of cloth. "Dividing" it would mean no more than taking half the threads.

Rashi says something very hard to read here:
"Since the mitzvah is with the goat, to push it over backwards... if you would say to tie it all to the rock, sometimes it would turn white early, before the goat was pushed. And he would be so heartened from the joy of the whitening of their sins, he would no longer care about doing the mitzvah of pushing the goat. Therefore, he ties it last between its horns, and his eyes are still on it. And since he is still dealing with the goat at the last, he won't fail to push it over. And the strip will not turn white at the halfway point (tied to the rock, not yet tied to the goat) until he has completed the whole mitzvah (of tying it)."
I've made some guesses at the translation, but I think this is a reasonable presentation.
But a number of things still need explanation:
1) Why does the lashon turn white before the goat is pushed?
2) Why won't it turn white until "the mitzvah of tying" is completed?
3) What is the "mitzvah of tying" anyhow? They used to tie it somewhere else (the heichal) and anyhow is this an avodah or just a siman or something?

Some suggestions on my questions - still needs a lot of work.
a) The forgiveness of Israel can happen before the goat is pushed. The gemara even says that the goat can die in the azarah, sending it isn't even m'akev, only hagralah is m'akev. So the forgiveness can be earlier.
b) However, the lashon doesn't turn white on its own. It will only turn white when it has become batel to part of the mitzvah of the sa'ir. That is, when it has been tied to the rock, or tied to the goat (or to the beis hamikdash itself in early history). Once it has been finished being tied on, it can become an analog measuring device of the forgiveness of Israel and turn white.
c) As I've suggested elsewhere, this notion of analog measuring device is also seen in the idea of the goral, of casting lots. In our day, casting lots is seen as random chance - by definition it gives no information. In their day they understood that it was a way of measuring kochos of tum'ah and taharah, of somehow comparing conflicting forces.
It could actually be used even by someone who wasn't even a servant of Hashem, such as Haman.