When the Roman Empire ultimately fell and the enemy breached the gate, what happened to the city? What did the sacking of a city in this time period entail? And for Rome did this affect the Christian seat at all, or was the fall of Rome long before it was the seat of the Christian faith?
In August of 410 CE, the Gothic king Alaric and his "barbarian" forces entered Rome by force upon the conclusion of a third siege. The entry point was the Porta Salaria, at the northern point of the Aurelian walls, but it is not known for sure if the gate was betrayed from within, or if the citizens capitulated out of desperation. It was the first time enemies had entered the city since 390 BCE, eight centuries earlier.
The human cost was massive. Although Alaric and his forces did not exactly engage in the widespread slaughter typical of the era, there was indiscriminate killing all the same, and rape, and all the rest. The wealthy were tortured into revealing the hiding places of their valuables. Most of those who survived were displaced as refugees. Jerome talks about refugee groups getting sold into sexual slavery in Eastern markets. As far as infrastructure, this first sack was not devastating. Structurally speaking, most of the buildings survived intact, including, it seems, the aqueducts which were still operational by that time. Their destruction would come later, in the several sackings still to come. The churches were largely left alone and allowed to offer sanctuary for terrified citizens lucky enough to make it there. Rome was still a populous place in 410, though probably not the city of one million it had once been. Its existence depended on a precarious balance of food shipments and water delivery and maintenance and interlocking industries, and most of these systems were disrupted in this event. I don't think we have exact numbers, but it is probable that the population declined sharply during this event and in the following years, despite the perceived lack of mass killing
But the riches of the city were not spared, and that included the adornment of nearly all the great buildings except the major churches. Many of the pagan statues in the city had already been melted down in attempts to pay Alaric's ransoms; now the remaining gold and silver objects of the city were stripped. The marble facades of the massive structures like the Baths of Caracalla were torn away; the Mausoleum of Augustus was destroyed, and the ashes of the Julio-Claudian family were scattered; Hadrian's mausoleum suffered a similar fate; the Gardens of Sallust were burned, as was the grove consecrated to Gaius and Lucius across the river. The destruction was deliberate and thorough. I have heard a late antique scholar claim that much of the decorative marble was, after three days of looting, carried south into Campania, and there abandoned because it was too heavy, possibly during the siege and sack of Nola and then Capua. It supposedly found its way into local Campanian noble houses in the medieval period, and thence back to Rome per the material demands of later Popes. Alaric would not survive for long after the sack, and legend holds that he was buried at a secret tomb in Italia which the Goths stuffed with riches, as was their way. Both the marbles and the tomb treasure are fun but impossible to prove.
The symbolic impact of the sack of 410 cannot be overstated. It was emblematic of the "fall of the West," even though the seats of Roman power had long since passed to the north, to Ravenna and Trier. Rome was the ancient capital, where the ancient senate sat and where pagan and Christian leaders ruled. The great structures of the capital had always been a physical representation of Rome's former glory and a promise of its return. The city would continue to be inhabited and resettled after the sack, only to be sacked several more times, but the glory of the capital never really recovered from this event. Though not yet ruins, places like the Mausoleums and the great Baths had lost their majesty and would steadily decline after this, quickly becoming quarries for materials rather than functional and symbolic sites.
A great source to start with is Peter Heather, The Fall of the Roman Empire (Oxford 2006). Also fun is Moorhead and Stuttard, AD410: The Year that Shook Rome (British Museum 2010).