After Rome collapses there is a pretty substantial amount of evidence to suggest that Roman structures survived past the collapse of the empire. Indeed there is some evidence that some of these practices, relating to tax collection, lasted down into the Carolingian period (800ish). The article "The Statistical Sources of Frankish History" by James Westfall Thompson highlights some of the practices that would have taken place in Francia that carried over from the Roman period.

As far as the cities are concerned that would depend on the city, however Ravenna (which was a Roman capital) retained a lot of its standing and would later become the seat of the Byzantine Exarchate. So I guess it would depend on what region and city you were looking at, but overall the bent amongst historians I have read seems to suggest more continuity than "total collapse" of the Roman world. Another example of a Roman city outside of Italy would be Tours, which still retained a lot of its Roman culture down into the time of Gregory of Tours writing his Histories. A further example would be Metz, which was also a prominent Roman town that was later influential to the Merovingians as it was a primary residence for the Frankish king of Austrasia.

For some suggestions of reading I would suggest Ian Wood's The Merovingian Kingdoms 450-751 which talks a pretty fair amount about continuity between late antique and early medieval Gaul. Pretty much everything above comes from Wood's book.

Speaking more broadly then I wouldn't necessarily view the fall of the empire as denoting a huge rupture of the other centers of Roman power, in many instances it seems that the former Roman cities essentially just accepted whoever the ruler was, as in instead of paying taxes to Rome, you pay taxes to the Franks. Hopefully this was what you were looking for!