Is it enough to sort for polynomially many 0-1 sequences for a sorting network?
It seems not.  Ian Parberry makes reference to a paper by Chung and Ravikumar, where they supposedly give a recursive construction of a sorting network that sorts every bitstring but one, and further deduce that the problem of verifying a sorting network is $co$-$NP$ complete.  I can't find the original paper right away, but certainly it matches (my) intuition.

Edit to add: It is actually very easy to find such a network that misses exactly one string.  The string to be missed will be $(1,0,\ldots,0)$.  Now you just want a circuit that sorts the last $n-1$ bits, then sorts the first $n-1$ bits.  This circuit will sort anything with at least two $1$s, will obviously sort the all-zero string, and will sort any string starting with $0$.