How does the auxiliary (AUX) power input behave on the 7055 series?

The external supply is only used when the downstream device actually draws power. If your DAC has its own power supply, you can leave the AUX input unconnected.
The AUX input ground is common with the isolated side so that power can be injected there. Any supply that covers the device's need is fine, typically below 0.5 A. A supply with a higher rating, for example 2.4 A, is safe; only the current that is needed flows.

An internal relay switches the AUX supply off when the host USB power disappears. You do not need an extra switch for the isolated side.

A linear power supply can lower the noise compared with the internal supply.
A well-designed DAC should, however, perform properly on any compliant
supply.

iPhones and many Android phones limit the current on their USB port. This is not enough to run the isolator with a bus-powered device. iPads and MacBooks supply enough current.