If I eat my nails/skin of my fingers or mucus on a fast it’s like eating?
It’s permited to do it not on a fast?
None of these are considered food, and it will be considered as if you fasted.
As a side point it is not a good idea to get into the habit of biting your nails or skin, because when you get into the habit, it will inadvertently continue on shabbos, which is forbidden. Another consideration; when a person bites his nail off, it is like he cut it off, and he now has to wash his hands. When he bites the second nail off he is putting his tameh hands into his mouth, which is not a good idea. This would not apply to biting off loose pieces of skin, as we don’t have to wash our hands after cutting off loose pieces of skin.
Regarding cutting some of the nails- It is brought in Orchos Rabeinu 3 pg. 186 that one doe not Hve to wash their hands after biting one nail, however see Tehila L’Dovid O:CH 4-17, Kaf Hachaim 4-68 in the name of a number of Poskim, that it also applies to cutting part of one’s nails. Also see piskei Teshuvos 4- ftnt 109.
Regarding Ruach Rah See Torah Lishmah 103. Also see Kaf Hachaim O:CH 4-61 from the Chida and Soles Blulah that the reason for washing our hands after cutting nails is because of Ruach Rah.