Gemara

(Mishnah - Beis Shamai): One who slaughters a Chayah or bird on Yom Tov...

Inference: This connotes that l'Chatchilah it is forbidden.

Contradiction (Mishnah - Beis Hillel): One may not slaughter (unless he has Muchan dirt).

This connotes that the first Tana (Beis Shamai) permits to slaughter!

Answer #1: Beis Hillel mean that one may not do a full Shechitah, i.e. Shechitah and Kisuy, unless he has Muchan dirt.

Objection (Seifa): They agree that if one slaughtered, he digs and covers.

This connotes that in the Reisha Beis Shamai permit Shechitah l'Chatchilah!

Answer #2 (Rabah): The Mishnah discusses one who asks what to do. Beis Shamai say, we tell him to slaughter, dig and cover. Beis Hillel say, we tell him that he may not slaughter unless he has Muchan dirt.

Answer #3 (Rav Yosef): The Mishnah discusses one who asks what to do after he slaughtered. Beis Shamai say, we tell him to dig and cover. Beis Hillel say, he may not slaughter unless he has Muchan dirt.

Suggestion (Abaye): Rabah and Rav Yosef argue about R. Zeira's law:

R. Zeira: "V'Shofach Es Damo v'Chisahu be'Afar" - this teaches that dirt must be put below (where the blood will fall) and above.

Rav Yosef holds like R. Zeira, therefore he requires digging beforehand. Rabah disagrees with R. Zeira, therefore he says to dig afterwards.

Rejection (Rav Yosef): No, Rabah and I both hold like R. Zeira. Rabah forbids if he does not have dirt ready for underneath. He is concerned lest the person decide not to slaughter after digging (so he dug on Yom Tov for no reason). I hold that it is better to permit him, lest he neglect Simchas Yom Tov.

10a - Question (Mishnah - Beis Shamai): On Yom Tov one may not remove Trisim (doors of store windows, to set up merchandise on them).

Beis Hillel permit even to return them. (Beis Hillel are more lenient for the sake of Simchas Yom Tov.)

Contradiction (Mishnah - Beis Shamai): One who slaughters a Chayah or bird on Yom Tov... (we established that they argue about one who lacks Muchan dirt who asks whether or not he may slaughter. Beis Shamai permit, and Beis Hillel forbid!)

We understand Beis Shamai. They are lenient about Shechitah when there is a shovel inserted in loose dirt, but there is no reason to permit removing Trisim (the Melachah of Stirah (destroying) applies to Kelim).

Why are Beis Hillel lenient about Trisim but stringent about Shechitah?!

Answer (R. Yochanan): Muchlefes ha'Shitah (the opinions must be switched).

Answer #2: Beis Hillel hold that Binyan (building) and Stirah do not apply to Kelim. (This line does not appear in the texts of some Rishonim. In some texts it is a rejection of R. Yochanan.)

Rishonim

The Rif brings the Mishnah verbatim.

Rebuttal (Ba'al ha'Ma'or): The Rif permits even returning Trisim, like Beis Hillel. If so, how can he bring our text of the Mishnah of Shechitah of a Chayah or bird on Yom Tov? R. Yochanan taught that the opinions must be switched (in one of these Mishnayos)!

Defense #1 (of Rif - Milchamos Hash-m): All the Ge'onim rule like this. R. Yochanan holds that Beis Shamai and Beis Hillel argue about a Tris in which the hinge is on the side. Returning it is a Melachah. However, it is only mid'Rabanan because mid'Oraisa Binyan and Stirah do not apply to Kelim. Digging for Kisuy is a Melachah for which one is exempt because of intent, i.e. he needs only the dirt. The law should be the same in both cases. Therefore R. Yochanan was forced to switch the opinions. However, we hold that they argue about a Tris in which the hinge is in the middle. According to letter of the law, one may remove it even on Shabbos. Beis Shamai decree to forbid even on Yom Tov lest one do so when the hinge is on the side. Beis Hillel forbid when the hinge is on the side lest one bang it in. Intrinsically it is permitted, because Binyan and Stirah do not apply to Kelim. Therefore they do not make a Gezeirah li'Gzeirah to forbid when the hinge is in the middle when this impedes Simchas Yom Tov. There is no need to switch the opinions. Some say that R. Yochanan himself holds that Binyan and Stirah apply to Kelim, but the Halachah does not follow him. I later found an accurate Sefardi text in which the Gemara rejects R. Yochanan for this reason.

Defense #2 (of Rif - R. Yosef, brought in Rosh 1:9): The Gemara explained why Beis Shamai are stringent about Trisim. 'Muchlefes ha'Shitah' means that Beis Hillel are lenient about Trisim, unlike their opinion in other matters of Simchas Yom Tov. This is because removing Trisim is a need of the Rabim, and the Tzibur cannot be stringent. The Yerushalmi says that if it is forbidden to return Trisim, people will not remove them, and this will reduce Simchas Yom Tov.

Rashi (9b DH Muchlefes): R. Yochanan switches the opinions here (regarding a ladder).

Question #1: The Sugya brought Mishnah after Mishnah, and each time suggested that R. Yochanan said that the opinions must be switched to resolve it with the Mishnah of Shechitah. Presumably, Rashi similarly explains that each time, including the conclusion, R. Yochanan leaves the Mishnah of Shechitah intact. If so, the argument of Rabah and Rav Yosef is according to Beis Shamai!

Question #2 (Tosfos 9b DH Amar): Each time (except for the last) the Gemara answered that the opinions need not be switched. These are poor answers. It is difficult to say that we must rely on them!

Answer to both questions (R. Tam in Tosfos, ibid.): The opinions must be switched in the Mishnah of Shechitah.

Answer #2 (to Question #1 - Milchamos Hash-m): The argument of Rabah and Rav Yosef is relevant also according to Beis Hillel. Rabah holds that Beis Shamai never allow digging before Shechitah; he holds that Beis Hillel never permit. Rav Yosef holds that Beis Shamai and Beis Hillel argue only about digging before Shechitah, but all agree that if there is dirt underneath, one may slaughter, and dig afterwards for Kisuy ha'Dam. The Halachah follows Rabah against Rav Yosef, according to Beis Hillel.

Rambam (Hilchos Yom Tov 2:18): If one had a shovel inserted in dirt from Erev Yom Tov and it came out on Yom Tov:

If loose dirt came out with it, the dirt may be used for covering, and it is not Muktzah;

If a clump of dirt came out, one may not grind it on Yom Tov.

Rambam (ibid. 3:1): If one has Muchan dirt or ashes that may be moved, he may slaughter a Chayah or bird and cover the blood. If he does not have such dirt or ashes, he may not slaughter. If he transgressed and slaughtered, he does not cover the blood until evening.

Magid Mishnah (2:18): The Rambam does not allow one to slaughter with intent to remove an inserted shovel and use the dirt that comes out for covering.

Rebuttal (Lechem Mishneh): The Magid Mishnah connotes that if one slaughtered, the Rambam permits to dig and cover. This is not so. He allows only if the shovel came out. Perhaps the Magid Mishnah explains that if the shovel came out one may slaughter l'Chatchilah, but if one slaughtered it is always permitted to dig. However, the Rambam (Halachah 3:1) connotes that one may never dig to cover.

Hagahos Maimoniyos (1): According to R. Tam, we switch the opinions in the Mishnah of Shechitah, so one need not prepare dirt from before Yom Tov for covering.

Rosh (1:9): Beis Hillel hold that b'Di'eved, if one slaughtered he may do Kisuy. Beis Shamai permit Shechitah l'Chatchilah. According to R. Tam, 'the opinions must be switched' (9b) refers to our Mishnah, so Beis Hillel permit Shechitah l'Chatchilah. Loose dirt suffices; Muchan dirt is not needed. Recently, people began to be stringent and require prepared dirt. This is unlike the Halachah. The Ri disagrees. Even though the Mishnah discusses loose dirt, we require that the shovel was inserted from before Yom Tov.

Poskim

Shulchan Aruch (OC 498:14): One may not slaughter a Chayah or bird on Yom Tov unless he has dirt prepared from before Yom Tov. If he transgressed and slaughtered, if he has a shovel inserted in loose dirt he may cover with it.

Beis Yosef (DH Kosav SMaG): SMaG, SMaK and Rabbeinu Yehudah say that if one slaughtered and does not have prepared dirt, he may cover with unprepared dirt. They rely on R. Tam.

Rebuttal (Yam Shel Shlomo 1:21): The Beis Yosef erred. If they held like R. Tam, they would permit l'Chatchilah even without an inserted shovel! If they are concerned for the Ri's opinion, even b'Di'eved they would require an inserted shovel!

Defense (Taz 12): They are unsure whom the Halachah follows. L'Chatchilah they are stringent like Rashi and the Rif. In pressed circumstances one may rely on them to avoid Bitul of Simchas Yom Tov.

Bach (DH ul'Inyan) Even Avi ha'Ezri says that the Halachah follows R. Tam (to permit when the shovel was not inserted for the sake of Kisuy), but in practice one should be stringent.

Kaf ha'Chayim (113): Some say that verbal designation suffices only for the dirt below the blood, but one must prepare the dirt for above through an action. L'Chatchilah one should fulfill this opinion. B'Di'eved, one may rely on those who say that verbal designation suffices even for the dirt above.

Kaf ha'Chayim (114): The dirt must be loose. Grinding dirt is a Toldah of Tochen, and it is forbidden on Yom Tov. Also, one must remove the dirt in a way that does not make a cavity.

Rema: It is permitted even if he must insert it many times.

Mishnah Brurah (75): The Rema holds that one who inserts a shovel before Yom Tov plans to use as much dirt as he will need. Some say that he intends only for the dirt on the shovel, and some say that the concern is digging on Yom Tov. Surely they forbid to reinsert it to take more dirt.