After looking at the logs you provided it looks like both your Hard Drive and Battery are getting ready to fail. The error codes just identify which of the tests registered a failure condition.
I would suggest that you backup all your documents that you can't afford to lose, and replace the hard drive as soon as you are able. Dell should be able to help you with this, and they would also be the best source for a replacement battery.
As for the issue of your system shutting down after 20 minutes, it is possible that your computer is pulling more power than is available. By my calculations, if your battery used to last for 5 hours the 6% of expected capacity would leave you just under 20 minutes on battery power. You could try removing the battery and running on just the power provided by your AC adapter to see if that makes a difference.
I hope that this helps, but please let me know if you have any other questions.
Thanks,
Kris