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When the Kindle is in sleep mode, it can index content. That uses a tiny amount of battery, but once the content is indexed, no further battery power should be used.
If your Kindle has Whispernet turned on, it will normally update itself once its "turn" has come. Amazon updates in batches, possibly based on serial number. There is a menu option for updating the Kindle, also, but it will not be active unless your Kindle is eligible. I believe that there are instructions on the Amazon web site for how to manually update the firmware.
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