No, because the DX3 Pro does not have analog audio inputs.
It's easy to switch between saved EQ profiles. There are two custom settings buttons on the remote that you can use to save EQ profiles and volume levels.
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Yes. Press the joystick to open the device menu. Scroll down to the menu line "9. PEQ setting" and "10. PEQ project", pressing left or right will turn on and off, and switch between saved (up to 5) EQ profiles respectively.
Impossible. See #1.
Direct import from AutoEQ is not possible. You will have to do this manually by rotating the frequency band sliders and saving the settings. See Topping Tune
reference guide.
The RME ADI-2 DAC FS can do all of this and much more. Plus a built-in headphone amplifier and a much more informative display.
The ADI-2 Pro FS R Black Edition and ADI-2/4 Pro SE can handle 2 independent headphone connections and EQ profiles, and also feature balanced headphone connections, analog inputs, and all of the above.