Does hashing require non-linearity?
Does hashing require non-linear components as well?


Yes


  How would a hash built from a linear psuedo-random permutation be vulnerable to collision/preimage search?


You could find a preimage by solving for the preimage with linear equations; that is, for a linear function, each output bit is a linear function of the input bits.  We would express the entire output as a set of linear equations (with the preimage bits being the unknown), set the right side to be image we're looking for, and solve that set of linear equations, giving us the preimage.