(Mishnah): Beis Shamai say that one may not take a pestle to chop meat on it (as it is usually used for grinding grain). Beis Hillel permit it.

Beis Shamai say that one may not lay down an animal hide where people are walking (in order that their trampling will work the hide), nor may he lift it after skinning unless there is meat on the hide. Beis Hillel permit it.

Beis Shamai say that one may not remove the boards from in front of a stall on Yom Tov. Beis Hillel permit even replacing it.

(Gemara): The Mishnah only permitted moving the pestle to chop meat, but if not for that use, it is prohibited (even according to Beis Hillel).

This follows the statement of R. Chinana quoting Rav - The Chachamim agree with R. Nechemya that Bezaiyra, Mezurah and Buchnah are prohibited -

(What are they?) A Bezaiyra is used for collecting and pickling; a Mezurah is a utensil used to bang clothes that are being laundered; a Buchnah is the peg used to comb silk fibers. (All of these utensils are set aside and the owner is very particular about them - this is the category called Muktzah Machamas Chisaron Kis and normally cannot be moved at all.)

The Mishnah had taught: Beis Shamai say that one cannot put an animal hide...

They both agree that he may not scrape it - since people will be trampling on it, it will not become ruined; therefore, there is no permission to work it.

Question: What is the case when the Mishnah says "unless there is meat on [the hide]''?

If the flesh is attached to the hide, the hide is considered part of the flesh.

If it has separated from the hide, it is like the statement of R. Yaakov bar Acha/ Chinana Kartachya quoting R. Hoshiya - A sack of money may be moved by placing a loaf on it and moving them together.

This is like the case of an Antichi (a copper pot that has two bases with the water in the upper section and the coals in the lower section) that fell on R. Yirmiyah on Shabbos. He put a pomegranate on top of it and moved it. It was not because the law follows R. Hoshiya (who permits this); rather, it was to prevent danger to R. Yirmiyah.

They both agree (even Beis Hillel) that he may not actually salt the skin.

Beraisa: However, he may apply a small amount of salt that is enough for roasting.

Chevraya quoting Rav: One may salt a large piece in one act, even though he is only able to eat a small amount (since he is only doing one act of salting).

R. Acha quoting Rav: One may salt (as if he wishes to eat it today) and deceive, then salt (another piece) and deceive - salt a piece over one part of its skin and then salt over another part of the skin until he has salted the whole skin. (This is permitted to prevent the financial loss that would have resulted from celebrating the festival.)