What is a strongly correlated system (in condensed matter physics)?

(1) Your definition of strongly correlated system is correct "single-particle fails." We can still use ARPES to study strong correlated systems, we just do not see features that would be present in a weakly correlated system. The most prominent feature in a weakly correlated system is a sharp peak at certain energy and momentum. If you track this peak in energy as a function of momentum using ARPES you have essentially measured the energy band. In strongly correlated systems this peak is not sharp. The precise definition of sharp is that it has a delta-function component to it (at least in theory).

(2) BCS SC is not a strongly correlated system. There are still energy bands in a BCS SC it is just that the energy bands do not describe electrons. The energy bands of a BCS SC tell you something about the SC quasiparticles called Bogoliubov quasiparticles. One interesting thing about Bogoliubov quasiparticles is that they carry non-integer charge.

(3) The "strongly correlated" refers to the interacting nature of the system and the fact that there is no single-particle description. If you excite a strongly correlated system in two steps, the excitation you make in the first step will effect which excitations you can make in the second step in a highly non-trivial way. The excitation you add first has a strong influence on the system and rearranges everything. In contrast, in a band metal you can add an electron at momentum $\mathbf{k}$ and the bands do not shift. You can then add another electron at momentum $\mathbf{k}'$ and its properties can be understood in terms of the original bands, i.e., the bands that were there before you added the electron at momentum $\mathbf{k}$.