1) THE REASON FOR HALF A SHEKEL
QUESTIONS: According to one opinion, the half-Shekel is given as atonement for the brothers' sale of Yosef. The ten brothers sold Yosef for 20 Dinarim, and thus each brother received two Dinarim, which are equivalent to one half-Shekel. All of the descendants of the ten tribes are obligated to give a half-Shekel each year to the Beis ha'Mikdash as an act of atonement for the sin of their forebears.
(a) If the Machatzis ha'Shekel is an atonement for the sale of Yosef, why is the money given to the Beis ha'Mikdash? It should be given to Yosef (or to his descendants)!
According to the explanation of the VILNA GA'ON (as recorded by the TIKLIN CHADTIN), it is clear why the money is not given to Yosef. The Vilna Ga'on explains that Yosef also acted wrongly by instigating his brothers' wrath against him, and thus he also bears responsibility for his sale. Of the twelve brothers, only ten were responsible for the sale of Yosef, including Yosef himself. Reuven and Binyamin were not involved in the sale of Yosef.
This explains why the money is not given to Yosef. Why, though, is the money given specifically to the Beis ha'Mikdash?
(b) Furthermore, since Reuven and Binyamin did not participate in the sale of Yosef, why are their descendants obligated to give the Machatzis ha'Shekel?
(a) The RAMBAM (in Moreh Nevuchim 3:46; see also Yalkut Shimoni #521) writes that the goats (Se'irim) brought as public sin-offerings on Rosh Chodesh and on Yom Tov, as well as many other Korbanos, are intended to serve as an atonement for the sale of Yosef. (This is because the brothers slaughtered a goat in order to use its blood to give the appearance that Yosef had been killed (Bereishis 37:31).)
Perhaps this explains why the half-Shekel is given to the Beis ha'Mikdash in order to attain atonement for the sin of the sale of Yosef. The half-Shekel is given to the Beis ha'Mikdash for the purpose of purchasing the Korbanos which atone for that sin.
(b) The SHEYAREI KORBAN on the Yerushalmi explains why the descendants of Reuven and Binyamin are obligated to give the Machatzis ha'Shekel. When the Gemara says that the half-Shekel serves as atonement for the sale of Yosef, it is not giving a reason for the Mitzvah to give Shekalim to the Beis ha'Mikdash; the purpose of the Mitzvah is for the sake of purchasing Korbanos. Rather, the Gemara's intention is to give a reason for why the amount that must be given is half a Shekel. Since most of the tribes sinned with the amount of one half-Shekel, it was designated as the amount that each tribe must give.
2) ONE WHO SLAUGHTERS THE "MOSAR PESACH" WITH INTENT TO MAKE IT AN "OLAH"
QUESTION: The Mishnah teaches that "Mosar Pesach l'Shelamim" -- a Korban that was designated as a Korban Pesach but was not offered at the appropriate time becomes a normal Korban Shelamim. The Gemara says that the Mosar Pesach is a valid Shelamim only if it was slaughtered l'Shem Shelamim, with specific intent that it be offered as a Korban Shelamim, but not if it was slaughtered with intent to be a Korban Pesach.
The Amora'im argue whether the Mosar Pesach becomes a valid Shelamim if it was slaughtered with intent to be an Olah. According to one opinion, the Korban is invalid and may not be brought at all, even as a Shelamim. According to the other opinion (Rebbi Yochanan), it is a valid Shelamim.
The Gemara then discusses the Halachah of a Mosar Pesach that was slaughtered with intent that it be an "Olah with a Machshavah of Pesul." That is, not only did he slaughter the animal to be an Olah, but he slaughtered it to be an invalid Olah. The Gemara asks whether such an animal will be a valid Shelamim or not. The question is left unanswered.
What is the Gemara's question? According to both of the previous opinions (with regard to slaughtering the Mosar Pesach l'Shem Olah), there should be no question. According to the first opinion, which says that the animal is not a valid Shelamim when slaughtered l'Shem Olah, it does not make a difference if he also slaughtered it with a Machshavah of Pesul, since it already became invalid when he slaughtered it l'Shem Olah.
According to the second opinion, which says that a Mosar Pesach slaughtered l'Shem Olah becomes a valid Shelamim, the Korban obviously will be disqualified when one slaughters it with a Machshavah of Pesul, an invalidating thought.
(a) The KORBAN HA'EDAH (as explained by the PNEI ZAKEN) explains that the Gemara's question applies only to the second opinion. The Gemara refers to a type of Machshavah of Pesul which constitutes Pigul only for an animal designated as an Olah, but not for an animal designated as a Shelamim. The Gemara asks what the Halachah is when a person has in mind this type of Machshavah of Pesul when he slaughters the Mosar Pesach to be an Olah. For example, his intent to offer the flesh of the Korban on the following day causes a Korban Olah to become Pigul, because the flesh is part of what is offered on the Mizbe'ach on the day that it is slaughtered, and thus intent to offer it after its proper time constitutes Pigul. In contrast, such a Machshavah does not disqualify a Korban Shelamim, because its flesh is not offered today on the Mizbe'ach. Rather, its flesh is supposed to be eaten tomorrow, and thus one who burns it, or has intent to burn it, tomorrow does not invalidate it at all.
Accordingly, is the animal considered Pigul because he slaughtered it l'Shem Olah, or is it not considered Pigul because it does not actually become an Olah, even though he slaughtered it l'Shem Olah? This is the Gemara's question.
This is what the Gemara means when it says, "Do you consider that it has been changed to an Olah with regard to a Machshavah of Pigul (that is, the type of Machshavah which invalidates only an Olah) [and thus the Korban will be disqualified], or do you not consider that it has been changed to an Olah with regard to a Machshavah of Pigul [and thus the Korban will not be disqualified]?" (This also appears to be the understanding of the TIKLIN CHADTIN in his initial explanation of the Gemara's question.)
(b) RABEINU MESHULAM and the TALMID SHEL RABEINU SHMUEL BAR SHNEUR explain that the Gemara here is merely repeating its previous question. The Gemara asked whether or not the Mosar Pesach becomes a valid Shelamim when it is slaughtered l'Shem Olah. The Gemara now gives another practical ramification and asks again, "Does a wrongful Machshavah during the Shechitah (such as intention that it be an Olah) disqualify it?" The practical difference will be in a case in which one has a Machshavah of Pigul. If the animal remains a valid Shelamim when slaughtered l'Shem Olah, then a Machshavah of Pigul will make it Pigul. If the animal does not become a valid Shelamim when slaughtered l'Shem Olah, then it is already Pasul and a Machshavah of Pigul will not affect it, because a Korban that is already Pasul cannot become Pigul.
RAV CHAIM KANIEVSKY takes this explanation one step further. Perhaps the Gemara, in its question here, follows the opinion of Rebbi Yochanan, who says that if one slaughters the Mosar Pesach l'Shem Olah, it is a valid Shelamim. The question applies in a case in which he slaughters it l'Shem Olas Pigul, to be an Olah but with a Machshavah of Pigul -- is his intention able to make the Korban into a Shelamim (albeit a Pasul one) or not? Perhaps only when he slaughters it l'Shem Olah Kesherah, to be a valid Olah, does it become a Shelamim. The Gemara then asks that in either case, the Korban is Pasul. The Gemara answers that if such a Machshavah makes the Korban into a Shelamim, then it will be Pigul. If such a Machshavah is unable to make it into a Shelamim, then it remains a disqualified Korban Pesach and it cannot be made Pigul.
(c) The TIKLIN CHADTIN has a different approach (as he explains in the Gemara's conclusion). He says that the Gemara follows the opinion of Rebbi Yochanan, who says that when the Mosar Pesach is slaughtered l'Shem Olah it becomes a valid Shelamim. The question is whether the Korban, since it was slaughtered l'Shem Olah, lacks a degree of completeness and is a valid Shelamim only b'Di'eved, or whether it remains a completely acceptable Shelamim, as though it was slaughtered l'Shem Shelamim.
The Gemara is asking whether a Machshavah to make the Korban into an Olah changes something in the Korban and makes it acceptable only "b'Di'eved" -- and thus it cannot become Pigul, because only a completely valid Korban can become Pigul, or whether such a Machshavah does not detract at all from the Korban and it remains completely valid and thus can become Pigul.