Who should handle customer support within an Agile team?

A customer support team or individual.

"Agile" isn't a method for running a business, it's a method for developing software.

That's not to say that a development team can also do customer support if the need arises, but that has nothing to do with being agile.

Who should handle customer support within an Agile team?

In general, a customer support team or individual should handle customer support. Proper customer support in the traditional sense of the word requires a distinct set of skills that many software developers don't possess. It's no different than who should handle the marketing or the finances or the janitorial services.

In short, "Agile" isn't a method for running a business, it's a method for developing software. You cannot look to agile to provide answers for how to run your business.

All that being said, in the purest form of agility this is a question only your team can decide. In general, some of the things you listed belong to the development team (tasks related to actual software development) and some to the product owner (interaction with the customer). None of the things you mentioned would belong to the scrum master.

If your team is tasked with every aspect of software development from initiation to final delivery and support, then it is your team's responsibility to figure out how to handle the non-software-delivery tasks since the solution will be unique to your company and your team.