Would I be able to keep an external hard drive next to my television, hook my Pandora up to both its charger and said external hard drive and be able to watch movies or play games off of the hard drive?
Yes.
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Or is the power draw just too much for the Pandora to handle
If you have something
like this that has (and requires) it's own power supply - you should totally be fine. You are talking about something immobile - a big TV - charging the Pandora, etc. - so if you get a big hard drive that needs its own power - you should be all set. I know some mobile devices can read flash drives - and powered hard drives - but not unpowered hard drives - (
like this) - because of the extra energy draw. I don't know if the Pandora will suffer from that same restriction - but we have been told external hard drives are doable.
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(I also heard that the Pandora can't just run off of the charger because of the way it manages power, is this also true?)?
I don't think so. But others have addressed this already.