Will core decomposition get a maximal clique?
No. The illustration shows a graph (the graph of a cube with one corner truncated) and a valid removal sequence such that the vertices left at the point when the minimum degree equals $|S|-1$ (the two red vertices) do not form a maximal clique.



In this example, some other valid removal orderings could still produce a maximal clique. Probably there are more elaborate versions of this example that force the remaining clique to be non-maximal for all valid removal orderings.