Is there a list of all normal subgroups for $S_N$?
What is a criteria for a finite group to be a normal subgroup of $S_N$?
Which of them are kernels of irreducible representation? From a partition of $N$, we can construct an irreducible representation, so how does the related subgroup look in terms of the partition?
It is a standard fact in group theory that when $N\geq 5$, the only normal subgroup is $A_N$, which in turn is simple. For $N\leq 4$, you can just do it by hand. There are is of course exactly two one irreducible representation whose kernel is $A_N$, the trivial and the sign representation.
I don't understand the last paragraph of your question, but there are lots of references on representations of symmetric groups, e.g. the representation theory book by Fulton and Harris.
