Rechargeable battery questions???

Factual Questions
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2 questions:

  1. My cellphone is charged by the cigarette lighter socket in my car when the ignition is on. My wife says I should leave the phone power button on 24 hrs a day…and I think the phone should be off during night when there is no charge…Is my wife correct?

  2. When my double AA rechargeable battery charger registers green lites showing charge is complete, am I shortening the life of those batteries by letting the fully charged batteries remain in the charger with the green lights on continuously? or should I pull the plug when they are fully charged?

Thanks for the info in advance. As you can see I am not very familiar with the proper terminolgy in my above questions.

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See http://boards.straightdope.com/sdmb/showthread.php?s=&threadid=129012