Seeing as everything is performed by a Kohen, why does the Torah repeat the word "ve'Lakach ha'Kohen"?
Targum Yonasan: To teach us that a third Kohen throws the three items into the fire.
Yoma, 43a: Because we would otherwise have thought that, since the three items are not part of the actual cow itself, they do not require a Kohen.
What is the definition of ?Eitz Erez??
Targum Yonasan: A log (or a piece) of cedar-wood.
What is the definition of "Eizov"?
Chulin, 62b: "Eizov" is a plain hyssop, not a Kuchli hyssop or a Roman hyssop or a desert hyssop or any other species of hyssop with a side name.
How much hyssop must the Kohen take?
Sifri: We learn via a Gezeirah Shavah "ve'Lakach" from "u'Lekachtem lachem Agudas Eizov" - in Bo, Sh'mos, 12:22 that he takes three stalks. 1
Since 'Agudah' always means at least three. Se Torah Temimah, note 32.
What is the significance of the three items that are thrown into the burning cow?
Rashi #1 (in Pasuk 22, citing R. Moshe ha'Darshan): They represent the three thousand who died [at the hands of the Levi'im] by the Eigel.
Rashi #2 (in Pasuk 22, citing R. Moshe ha'Darshan): The cedar is the tallest species of tree and the hyssop, the shortest. It shows that one whose vanity causes him to sin, should make himself small like a hyssop and like a worm, 1 and he will be forgiven.
Since Tola'as' means, both a 'thread' and a 'worm'. There is also a tradition that the cedar wood and the hyssop Are tied together with the red thread. See Oznayim la'Torah:
How will we reconcile the contradictory terms "S'reifas", implying that it is ashes, and "ha'Parah", implying that it is still intact/
Sifri: We learn from here that the Kohen throws them into the fire when the fire has caught hold of the majority of the cow - when it it is stil called a cow but is also already ashes.
Why does a Kohen add the three items to the burning cow?