Yes, with the CUDA collision solver (available in the beta, or PRO) you can have proper inter-cloth collisions. Otherwise you need to use Particle Physics.
Not sure what you mean by copying an instance of the cloth operator to another event...unless I'm misunderstanding what you're talking about, that wouldn't have an impact on inter-cloth collisions.
Not sure what you mean by copying an instance of the cloth operator to another event...unless I'm misunderstanding what you're talking about, that wouldn't have an impact on inter-cloth collisions.