Neither Rome nor Constantinople were 'destroyed.' Both have been sacked at different times in their history, and both have changed hands between differing national and cultural forces.
But both cities have been the focal point of a declining empire, whose desecration symbolically with the former and literally with the latter signalled an end to that empire. The fear of Gotham's decline is thus - intentionally or not - linked to the idea of an American
decline, the city's endemic corruption and crime emblematic of the broken social contract.
Myself, I find the notion of a fanatical, millenium-old apparently secular organization devoted to mass murder to be utterly at odds with virtually all of human history,
Yes, but this is a superhero movie. Thus, it's basically a pulp story. And this organization would have not been more believable if it had a religious origin, nor is it really much more preposterous than the ridiculous science or the idea that one man with unlimited funds could really turn himself into such an effortless one man army (and that this rich man's fantasy would seriously be the best way to combat crime).