What will be the behaviour of superconductors if they are used as core or coil?
As superconductors are perfectly diamagnetic, I would describe putting them in a transformer core as misuse, rather than use. They will do exactly what you don't want in a core material. Soft iron allows many times more flux for any given field than air, a superconductor allows zero flux, regardless of the field.

Superconducting windings, coils, behave just like ordinary windings, as far as having voltages induced on them due to changing magnetic flux linkage, and generating a field due to current flow through them, but without the \$I^2R\$ heat.