Let assume that we have two pairs of socks, N1 and N2.  Only two pairs, for simplicity.


Let the pair N1 is the pair  (1L,1R) : N1 left sock  and  N1 right sock.

Let the pair N2 is the pair  (2L,2R) : N2 left sock  and  N2 right sock.


We start from this arrangement

    1L,1R,    2L,2R.        (1)


All your interpretation is based on considering the pairs like

   (1L,1R),  (2L,2R)        (2)

and their derivatives.


To make derangement, I transpose sock 2R from the second pair to the first pair 
and transpose sock 1R from the first pair to the second.  I get

   (1L,2R),  (2L,1R).       (3)


In your interpretation, two pairs in (3) are deranged: you count it as deranged.


Let's write (3) as a row of socks  (without separating them in pairs)

    1L, 2R, 2L, 1R.


But now two socks 2R and 2L do belong to one pair and are next to each other, so this arrangement 
can not be counted as deranged, and the logical construction is destroyed.


So, your interpretation has an interior contradiction.


It is WHY I could not solve the problem and it is why I think, that the problem, 

as it is worded, printed, posted and presented, is DEFECTIVE and can not be solved in terms of derangement.