1) WHY DOES "HAZA'AH" NOT MAKE THE KOHEN GADOL "TAMEI"
The Mishnah states that during the seven days before Yom Kippur, the Kohen Gadol performs the Avodah in the Beis ha'Mikdash.
The Gemara asks why the Kohen Gadol may perform the Avodah during those seven days. The Kohen Gadol is required to have Haza'ah done to him on each of the seven days out of doubt that he might have become Tamei with Tum'as Mes prior to his seclusion. However, when Haza'ah is performed to a person who is actually Tahor, the Haza'ah makes him Tamei. If the Kohen Gadol indeed was Tahor before the Haza'ah, the Haza'ah makes him Tamei and thus he should not be permitted to perform the Avodah during the seven days of his Perishah. (If Haza'ah is not done to him, the Kohen Gadol may perform the Avodah because of a Chazakah ("Chezkas Taharah") that he is not Tamei with Tum'as Mes.)
In its first answer, the Gemara says that the Mishnah follows the opinion of the Rabanan who argue with Rebbi Akiva. The Rabanan maintain that when Haza'ah is done to a person who is Tahor, he does not become Tamei.
In its second answer, the Gemara says that the Mishnah may follow even the opinion of Rebbi Akiva, who says that Haza'ah done to one who is Tahor renders him Tamei. However, the Haza'ah done to the Kohen Gadol during his Perishah is done only in the evening, immediately before nightfall, after he has completed the Avodah for the day. He becomes Tamei only until nightfall, and thus he is fit to perform the Avodah the next day.
The OSTROVTZER GA'ON, Rav Yechiel Michal zt'l of Ostrovtz, suggests a different explanation for why the Kohen Gadol is permitted to perform the Avodah after Haza'ah has been done to him.
The Mishnah in Mikva'os (6:2, cited in Chagigah 22a) states that when a utensil, with a smaller utensil inside of it, is immersed in a Mikvah, the inner utensil becomes Tahor because the water that seeps into the outer utensil is considered part of the Mikvah. However, this applies only when the opening of the outer utensil is larger than "k'Shefoferes ha'Nod" (the size of the mouth of a flask). If the opening in the outer utensil is smaller than "k'Shefoferes ha'Nod," the water inside the outer utensil is not considered connected to the Mikvah, and the inner utensil does not become Tahor. Nevertheless, the Tevilah is effective even in such a case when one condition is fulfilled. If the outer utensil itself was Tamei, the Tevilah works for both the outer utensil and the inner one. Since the Tevilah makes the outer utensil Tahor, it also makes the inner one Tahor (though "Migu").
The Ostrovtzer Ga'on suggests that before the Haza'ah is done to the Kohen Gadol each day, the Kohen Gadol should touch a utensil which has a very small opening. A small utensil that is Tamei should be placed inside the utensil that the Kohen Gadol touched. The two utensils should then be immersed in the Mikvah.
If the Kohen Gadol was Tamei, then the larger utensil became Tamei when he touched it, and thus the Tevilah works to make the inner utensil Tahor. If he was Tahor, then the large utensil is also Tahor and the Tevilah is ineffective for the inner utensil, which thus remains Tamei. In short, the status of the inner utensil is the opposite of the status of the Kohen Gadol; if he is Tamei, then the inner utensil becomes Tahor, and if he is Tahor, then the inner utensil remains Tamei.
Next, the Mei Chatas (the mixture of the spring water and the ashes of the Parah Adumah) should be placed into the inner utensil. If the Kohen Gadol was Tamei until now, then the utensil is Tahor and the Mei Chatas is valid and serves to make him Tahor.
On the other hand, if the Kohen Gadol was Tahor, then the inner utensil is still Tamei (because its immersion while inside the larger utensil, which was Tahor, was not a valid Tevilah) and the Mei Chatas that is placed in it becomes Tamei as well. Mei Chatas which is Tamei is invalid for Haza'ah. Since the Mei Chatas is invalid for Haza'ah, it cannot be Metamei a person who is Tahor. Consequently, Haza'ah with this Mei Chatas may be done on the Kohen Gadol during the seven days of his Perishah, and he may perform the Avodah without concern that the Haza'ah might have made him Tamei, even according to Rebbi Akiva.
2) ONE WHO BECOMES TAMEI BY TOUCHING THE "MEI CHATAS"
QUESTION: Rebbi Akiva and the Chachamim disagree whether or not Haza'ah done to a person who is Tahor renders him Tamei. In the case of the Gemara here, Haza'ah is done to the Kohen Gadol out of doubt that he might have become Tamei with Tum'as Mes. If he is really Tahor, according to Rebbi Akiva the Haza'ah makes him Tamei, and according to the Chachamim he remains Tahor.
The Gemara later says that Rebbi Akiva and the Chachamim agree that one who touches or carries Mei Chatas becomes Tamei. (One who carries Mei Chatas becomes Tamei only when he carries an amount of water into which the Ezov wood can be dipped, and with which another Haza'ah can be done. One who touches Mei Chatas becomes Tamei even when the amount of water that he touches is not enough to perform another Haza'ah.)
If the Chachamim agree that one who touches or carries Mei Chatas becomes Tamei, then why do they say that Haza'ah done to one who is Tahor does not make him Tamei? In that case, the Mei Chatas touches him, and the Chachamim agree that one who touches Mei Chatas becomes Tamei. Although they maintain that the act of Haza'ah does not make him Tamei, the Chachamim should agree that he becomes Tamei when the Mei Chatas touches him.
(a) The ME'IRI explains that when the Mei Chatas is sprinkled on a person who is Tahor, it is not called "Maga" (touching). Apparently, the Me'iri understands that the quantity of water sprinkled on the Kohen Gadol is not enough to perform another Haza'ah with it (and thus his act of carrying the water on his body does not make him Tamei).
It is not clear why the Me'iri says that this is not called "Maga." At the moment of the actual Haza'ah, the Mei Chatas cannot make him Tamei; the Mei Chatas cannot be Metamei during the process of being sprinkled. It can be Metamei through contact only outside of the process of sprinkling (TOSFOS YESHANIM DH Rashi Gores). Even if the Kohen Gadol does not really need Haza'ah, the Mei Chatas is not Metamei him because it was done for the sake of Haza'ah. Why, though, is the Mei Chatas not Metamei him after the process of Haza'ah is completed and the water is touching his body?
(b) The RAMBAM (Hilchos Parah Adumah 15:1) writes that as soon as one immerses the Ezov for the Haza'ah, the Mei Chatas becomes "Mei Chatas she'Na'asah Mitzvasan," Mei Chatas which has already been used to perform the Mitzvah, and it no longer is able to be Metamei.
The Rambam implies that although the Kohen Gadol was actually Tahor, since the Haza'ah was already done the Mei Chatas does not make him Tamei through Maga, because it is Mei Chatas which has already been used to perform the Mitzvah.