Why does cutting out part of Orochimaru’s soul cause him to lose the use of his arms?
I've just re-watched the scene in the anime where that happened, and it looked to me like the death god was clearly pulling on the part of Orochimaru's spirit that controls his arms(you can see hands at the end).  That just happens to be where the death god was pulling on, and it happens to only have been pulled out to his elbows or so.  

Maybe the 3rd drew the arms out on purpose because he deemed that to be the easiest or most effective thing to pull out first in-case he only had to partially do the job.  

There's really not much more to it than that.