Firstly: Cutting the Nails
It is a sunnah  There is nothing authentically narrated from the Prophet ﷺ in terms of how to cut the nails. It is permissible to discard them anywhere, and burying them is not obligatory. to cut the nails, and this is by agreement of the four schools of jurisprudence.
Secondly: Washing the Knuckles
It is sunnah  Al-Nawawi said: “As for washing the knuckles, then there is agreement over it being mustahabb.” Al-Majmu` (288/1) to wash the knuckles and joints.  Barajim is the plural of burjumah, and it refers to the joints that section the fingers.