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Modern hard drive are designed to cope with a certain amount of bad blocks. I'd lean towards the latter, as if the drive failed a Hardware Test that isn't a good sign.
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Two options are, to create a USB installer for Mavericks and try reinstalling Mavericks on that same hard drive, or to buy another hard drive and install Mavericks using those instructions. If your MacBook isn't offering that option then I assume that means it didn't receive this update. Internet Recovery might be an option, but unfortunately for Internet Recovery to work on a 2010 MacBook it would have needed to have this early 2012 EFI Firmware ROM update installed. If you are unable to boot into OS X and unable to boot into the Recovery then it suggests both are corrupted in some way. If you installed Mavericks on a 2012 MacBook that drive should boot okay on a 2010 MacBook. The differing architecture shouldn't be a problem.
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