My limited understanding of history says that he was persuaded/bribed not to do so by the then-current pope. Would it have changed anything if he had taken the city, and perhaps Constantinople?
No, not exactly. The Pope did indeed ask Attila to withdraw, which he did.
However, the Hns were more dangerous because they ranged and raided fast. Militarily, they were relatively weak and their best efforts against the Romans were basically draws. They are mostly remembered for having burned Aquileia (the survivors then went down and founded Venice).
Doubtful. Rome had been sacked 40 years earlier by Alaric ( Visiogoth ) and would be sacked again just a couple of years after Attila’s foray into Italy, by Gaiseric ( Vandal ). In all it got the loot and burn treatment at least seven times throughout the ages.
Constantinople wasn’t quite impregnable, but it was a lot closer to it than Rome. I sincerely doubt Attila could have pulled it off without a fleet.
As smiling bandit noted, Attila was a precursor to the Magyars - he was after booty and bribes as a means to strengthen his hold over his steppe empire. He was not, unlike some of the Germanic invaders, looking to supplant or usurp Rome’s authority in Italy.