Who should handle customer support within an Agile team?
"Customer Support"? Seriously, customer support is handled by a "Customer Support" team. It needs wildly different skills than programming, having a developer do that is in the same league as have him be the janitor or accountant. Maybe a specific developer could, but generally speaking, chances are it's a really bad idea.
Responding to user bug reports & creating stories/issues for them
This is a product owners job. Receiving feedback from stakeholders and getting them into shape for package of work for the team is the product owners main job.
Responding to negative app store reviews
Respond how? As in marketing? As in community relations? That's different jobs. And they need a different skill set than developers normally have.
Troubleshooting user problems
That may or may not be the developers job. Most companies have a different team to do this, too, but at least, of all those points, this is one that a developer should be qualified to do.
Responding to overall user feedback & feature requests
Talking to stakeholders is a product owners job.
Fixing urgent defects that were not known at the time of sprint planning
After prioritization by the product owner, as soon as the defect task/story is in the sprint backlog, that indeed is a developer task.
Customer support is a job. It's not something a developer does in their lunch break. You need a different skill set, too. You need to hire somebody for that the same way you would hire an accountant. You'd never have a developer do the companies books one day a week either.