Does this use the battery? Not clear to me if the Kindle program is running continuously or if it just periodically starts up, refreshes the screen and then powers back down.
I've set up my own DIY version of this. It pulls a new image via a cron job and repaints the screen with it. I've not invested time looking into it but by default if the device sleeps, which it does very quickly after loosing wall power, the cron job will no longer run and the display gets stuck. You need to poke the power button to get it to wake again.
Instead of using a cronjob you can put the device into sleep and use the RTC to schedule the next wakeup (see [1]). This takes only very little power, as the device is only turned on for mere seconds and sleeps the remaining time.
I presume it does, but would only need charging once every couple of months. If it's going to be in a fixed position, e.g., on a desk, then it could just be left plugged in.