Explanation #1 (Rashi, in Avodah Zarah 7a): This refers to thieves and Amei ha'Aretz who repented from their evil.

Question #1: This is astounding! Why don't we accept them? It connotes that there is no solution for an Am ha'Aretz to become a Chaver in any way!

Question #2: Above, it teaches in R. Meir's words "an Am ha'Aretz who accepted matters of Chaverus..."

Question #3: "Olamis" (ever) connotes that they already accepted them another time!

Question #4: R. Yehudah says that if Chazru covertly, we do not accept them. If they repented covertly, but not in public, why do we not accept him, since he fears Shamayim covertly?!

Explanation #2 (R. Tam): It refers to the Reisha, which taught in the Tosefta in Demai (2:3) an Am ha'Aretz who accepted on himself the matters of Chaverus ... a convert who accepted on himself Divrei Torah... a Kohen who accepted on himself Kehunah... a Levi who accepted on himself the Avodah of Leviyah....

And then it teaches all of them that Chazru... i.e. they returned to their evil ways and corrupted their deeds, we never accept them, since they could not persist in it;

R. Yehudah says, covertly. I.e. they returned to their evil ways covertly, and in public they are careful about matters of Chaverus, we do not accept them, for they deceive people, and they do only for appearances;

However, if they totally ruined their ways even in public, we accept them when they repent, for when they repent, they totally repent.

Explanation #2 (cont.): Some say that if they did their deeds covertly, that we knew them initially before they returned to their evil ways covertly, that they were conducting matters of Chaverus even covertly, we accept them when they repent;

However, if we know that always, they conducted matters of Chaverus only in public, but covertly they were not careful, we do not accept them, for they deceive people.

Question: Tamei Terumah is not [ever] proper for him. Is one not believed [about Taharah of his Terumah], and similarly regarding other Isur v'Heter?!

We explicitly say in Eruvin (63a) that a Chacham can inspect his own [Shechitah knife]!

Implied question: Even though it explains the reason because the benefit is to the owner, you are forced to say that it is mid'Oraisa, like is proven in Temurah (8a), and in Sanhedrin (112b) regarding Ir ha'Nidachas;

It expounds "Behemtecha" - what is eaten like your animal. This excludes Bechor and Ma'aser, which are not eaten like your animal, like [our] the Mishnah teaches "all Pesulei ha'Mukdashim may be sold in the market, except for Bechor and Ma'aser!"

Note: The Gemara in Temurah cites our Mishnah, but the Gemara in Sanhedrin does not. Tosfos connotes that it is cited [also, or perhaps only] in Sanhedrin!

Answer: Perhaps we learn from some verse that we do not act disgracefully with something that Hekdesh does not benefit from.

Implied question: According to Rava, who permits an adult through Havla'ah (he pays one price for the skin, Chelev, Gidim and horns, and gets the meat for free with them), why did Rav Sheshes need to establish it to discuss orphans?

Answer: It connotes to him our Mishnah is without Havla'ah, similar to [selling] Pesulei ha'Mukdashim in the market.

Explanation: One may not sell the meat, only the Chelev, skin and sinews.

Inference: Even the one who forbids below (32b) to give [meat of a Ba'al Mum] Bechor to a Nochri, the Chelev is permitted, for it is considered like the skin, since it is forbidden for Yisrael to eat it.

Distinction: The fat of the Gid [ha'Nasheh], which is not forbidden mid'Oraisa, perhaps it is forbidden [to give to a Nochri].

Pesak: However, [this is not relevant l'Halachah, for] the Halachah follows R. Akiva, who permits even the meat to a Nochri, like I will explain below (32b DH Pesak).

Observation: In some places we are meticulous about this, e.g. in Bava Metzi'a (2a) regarding "this one says I found it..." [you cannot say that the Mishnah taught only one clause, for] it taught "this one... and this one..."!

And in Nedarim (4b) it asks 'why does it say "Asur" twice?'

And in Sanhedrin (3a) it asks 'why does it say "three" twice?'

And in Sanhedrin (8a) it says 'if you will say that it explains "what is Zimun? It is Birkas ha'Zimun", [this cannot be, for] it says Zimun is with three, and Birkas ha'Zimun is with three!'

And in Mo'ed Katan (22b), that it explains [the Isur of an Avel for] Simchah of friendship is 30 days, it asks "but a Beraisa teaches "for Simchah 30, and for friendship 30 days!"

And in Bava Basra (104a) it asks 'why does it say twice "I sell to you"?'

Question: In several places [the Gemara] does not ask so - in Chulin (16b) regarding one who inserted a knife in a wall and slaughtered with it, the Shechitah is Kesherah;

[The Gemara] says that Perushi ka'Mefaresh (the Seifa of the Beraisa explains the Reisha). It says "what is the case of something attached to the ground? It is an attached knife, which one is not Mevatel (leave it permanently in the wall)." It does not ask 'why does it say twice "his Shechitah is Kosher"?'

And in Chulin (98a) regarding Tahor eggs cooked with Tamei eggs, it says Perushi ka'Mefaresh, and it does not ask 'why does it say twice "they are forbidden"?'

And in Chulin (113a) regarding Tahor fish salted with Tamei eggs", it says Perushi ka'Mefaresh, and it does not ask 'why does it say twice "it is permitted"?'

And in Bava Kama (98b) regarding one who gave [materials or a Kli] to craftsmen, and they ruined it, it says Perushi ka'Mefaresh, and it does not ask 'why does it say twice "they must pay"?'

And in Gitin (42b) regarding "all my property is given to Ploni and Almoni my slaves", it says Perushi ka'Mefaresh, and it does not ask 'why does it say twice "they did not acquire"?'

Answer: In all of these places that it asks "why does it say twice?", it is not so applicable to say Perushi ka'Mefaresh like in the places where it does not ask.

Explanation: [He holds like Rava] due to a different verse, but not for Rava's reason.