What is the meaning of 'sacrificing one's evil inclination'?

Maharsha: One who overcomes his inclination going forward and confesses over the past is considered to have 'sacrificed' it. This can be used to explain the Gemara in Shabbos 129b, which states that on Erev Shavuos a spirit named "Tavuach" went out, such that if Yisrael would not accept the Torah, it would slaughter their "flesh and blood". One may ask: why is the spirit called "Tavuach" ('slaughtered') rather than "Toveach" ('slaughters')? Based on our Gemara, one who overcomes his evil inclination is considered to have slaughtered it. Since Yisrael accepted the Torah and crushed their evil inclination, the spirit itself was effectively 'slaughtered', and is therefore called Tavuach.

Ben Yehoyada: The evil inclination (Yetzer hara) is metaphorically the heart (Lev), which has a gematria of 32. Sacrificing it means breaking the heart into two parts and confessing over each part, representing the hidden and revealed aspects of life. Each part is associated with a double Vav-Yud (16+16), whose total numerical value also equals 32, echoing the gematria of the heart and corresponding to the 32 mystical paths of wisdom. By doing this, a person honors Hash-m in both the hidden and revealed worlds, turning his inner struggle into a spiritual offering (Korban).

Why did Hash-m tell him to cast lots, when either way, Hash-m would ensure that the outcome would incriminate the sinner? Why not just tell Yehoshua directly who the sinner is?

Sefer Chofetz Chaim (Rechilus Klal 10:2:6 w/ Be'er Mayim Chaim): We learn from here that even when it is necessary to reveal something negative, it should, whenever possible, be done in a way that minimizes the explicit verbalization of Lashon Hara.

What was the reason that Achan took from the spoils?

Ein Eliyahu: The Torah states (Devarim 20:14), "And you shall consume your enemy's spoils". Achan understood this to be a positive Mitzvah. On the other hand, the Midrash states (Tanchuma Naso 28; Yalkut Yehoshua 15) that Yehoshua decreed that it is prohibited to take the from the spoils of Yericho. Yehoshua said that since it was conquered on Shabbos, just as Shabbos is holy, so the spoils collected on Shabbos are holy, and may not be used by Am Yisrael. Achan held that the positive Mitzvah to use the spoils overrides this.

Why did Achan claim that his sin should not have caused Yisrael to be punished, despite the concept of Arvus - communal responsibility?

Arvei Nachal (Beha'aloscha Drush 2): Klal Yisrael are not responsible for a Jew who has excluded himself from the Klal because of his sins. Achan claimed that since he had desecrated Shabbos three times, he was no longer part of the Klal and he could therefore not be the cause of the death of the 36 men (as discussed in Yehoshua 7:5 and the Gemara on 44a)

How could Yehoshua say something to Achan that seems to be a lie, that if he would confess, he would be dismissed from punishment?

Chamra veChayei, Yaavetz: Yehoshua did not actually lie. When he said that Achan would be "dismissed from punishment," he meant that Achan would be exempt from punishment in the World to Come. Since Achan would receive the death penalty in this world, that punishment would serve as atonement for his sin. Yehoshua spoke in deliberately ambiguous language in order to encourage Achan to confess. Although Achan may have understood Yehoshua's words to mean that he would be spared punishment in this world, Yehoshua himself spoke truthfully, as he was referring to exemption from punishment in the World to Come.

According to R. Yehuda, how can Am Yisrael be held responsible for the sins of an individual that are hidden from anyone else?

Chamra veChayai (citing 'Klalei haMeiri'): We are not responsible for the hidden sins of a person who has a 'Chezkas Kashrus' ('presumed to be an observant Jew'). We are responsible only for the sins of one who transgresses openly, as if he sins publicly in minor matters and is rebuked, he may be prevented from committing more serious sins in private.

Meiri: R. Yehuda understands that Am Yisrael is obligated to investigate possible hidden sins. If this is not done sufficiently, they are held responsible. (Sanhedri Ketana adds that the requirement to investigate is only after we see that Hash-m punished the person.)

Rabbeinu Yonah: There will always be a discernible blemish in a sinner, so even "hidden" sins have some level of exposure that creates responsibility for others.

Maharalbach (Teshuva 77): Responsibility for hidden sins applied only in this case, which was unique: all of Klal Yisrael were gathered together under Hash-m's direct supervision and were experiencing miracles beyond nature. Because of this extraordinary situation, the community as a whole needed to be fully righteous. Even one person's hidden sin was like a disease in a single limb of the body, affecting the entire nation. In this way, the hidden sin of an individual weakened the community, making them unfit to receive Hash-m's close supervision.

Maharsha: The community were responsible here because the sin, though hidden, could have been prevented. Since Yehoshua voluntarily imposed the Cherem, he should have appointed guards to ensure it was kept, and the people were held liable as a result. According to R. Yehuda, communal responsibility applies only when hidden sins could have been prevented, not for hidden sins that could not have been detected.