I have a Garmin portable GPS device with European maps, which I'd like to use while driving. Can I reasonably expect European rental cars to have the power outlet to run my GPS, or do most travelers recharge each night and run off the battery?
European cars have the same outlet as American cars. You'll be able to use your regular power cord. In my experience, the battery won't last all day, anyway.
Same answer as Nancy, same power outlet in European rental cars as here. Not a problem.
Of course European cars have cigar lighters.
If you plan to use it while driving as you say, there is no concern for the battery. I leave my Garmin connected to the cigar lighter when I use it while driving.