Keep the MBPR unplugged when not in use

If you bought a new battery, and that didn't fix the issue then I would say it is a short in the logic board. If it is still under Apple Care, I'd take it into a store, or call 1(800)MY-APPLE to get it checked out, and serviced.

To get the most out of your MacBook's battery, follow the Notebook Battery advice from Apple: unplug and use your battery until empty about once a month, then charge back up to full. Your question expands a bit on another question on the site about general charging:

Should I disconnect my MacBook Pro's power cord when the battery is fully charged?

In your case, streaming video is usually a high power usage event that can drain a battery in 4 or 5 hours. You will therefore be more likely to have a higher recharge cycle count than someone that isn't using as much power. For that reason, I would try to keep it plugged in when running. Everything is a balance, and it would be hard to reach the 1000 recharge cycles design lifetime of the new batteries in Apple portables - but as long as you are once a month draining it down, you might be better off keeping it on power the rest of the time it runs just in case you reach that limit before you are done with using the Mac.