It depends on what these key words mean - "operate" on what level? and "fall" as in 476? A number of cities continued to provide public services, pay to keep up their existing infrastructure and construct new buildings. It depends on how you define the specific romanitas of a city! In Ostrogothic Italy, some aqueducts were repaired or built anew in Ravenna, Rome, Naples, for instance. Vandal North Africa also remained pretty urban. Overall, cities definitely get less important for a few hundred years, for sure.

Very interesting question. I honestly don't know.

If any kind of resistance was there, I think it was more of a cultural one rather than an organised armed one.

But it is definitely something that needs to be studied ! We never know and might be surprised

Yes, read here. https://www.labrujulaverde.com/en/2019/08/the-5-great-last-battles-of-the-western-roman-empire/