I somehow managed to lose a page of my Lovelace dashboard in that the entities are now all wrong on the cards. Is there a way to restore the dashboard configuration and thereby save me the considerable effort to figure out the correct entiies associated with 15 temperature sensors?
You’ll have to restore a backup of HA to get that. Hopefully you take them regularly! I take one every morning just in case stuff like this happens 
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Is that part of the backups that gets done when there is an update or is that a backup that must be explicitly taken of .storage, etc?
You should be doing both. And copying the backup off the system automatically.
EDIT: See option 5: https://www.home-assistant.io/common-tasks/os#copying-your-backups-to-another-location