Will I be able to use the same Quell Relief App for both the Quell Classic (QE-001, QE-002) and Quell 2.0 (QE-020)?

Yes, the same Quell Relief App works for both the Quell Classic and the Quell 2.0.  If you already have the app and purchase a new Quell 2.0 (QE-020) device, you will need to make sure you upgrade to the latest version of the app (3.0) which supports the new device.