Depends on the panel you use, and how you use it, e.g. partial refresh after hibernate.
You would avoid issues like here: https://forum.arduino.cc/t/gxepd2-loss-of-contrast-waveshare-e-paper-driver-hat-rev2-3/1190325/5.
Yes.
DESPI-C02 with the e-paper panel consumes very little current when set to deep sleep (hibernate).
And little current while not refreshing, less than the level converter of the WS HAT. See also specs of the panel.
The new HAT 2.3 from Waveshare will consume zero current when PWR is not pulled high, and all control and data lines are set floating, set INPUT. Call display.end(); after display.hibernate(); and before setting processor to deep sleep.
BTW: there is also a Good Display shop on AliExpress: https://goodisplay.aliexpress.com/store/600281
-jz-