I'd check https://github.com/Josh-writes/microslate-firmware/ for the xteink as it is less focused on only reading, but keep in mind it's a screen only, so not touch. Buttons though are pretty good so enough for tick items on a list IMHO.

Honestly it's pretty good, but it's too new to give you an actual number. I haven't had to think about it so far. As a LoRa Meshtastic node the device ran for days and still had plenty of battery.

In normal operation battery life is by how often you do a refresh, the rest of the time it's in deep sleep. I think for this application I could build in a refresh schedule tied to when the house is active (mornings, evenings, weekends) to reduce the number of refreshes. It syncs on touch if it hasn't synced in the last 5 minutes (and there's a button for when your feeling impatient)