Unclear diagnosis of hard drive failure

It sounds to me like your file system has become corrupted. This means that the hardware is most likely fine, but the data stored on it is so currupted that the only solution is to boot to your recovery partition, run the Disk Utility to re-partition the drive & format it, then re-install your MacOS on it. Once that is done, you can restore from your backups.

Since this will be wiping the HD that has your recovery partition, you may lose that as well so if you can do so, I suggest that you make sure to download the Recovery Disk Maker from Apple and build yourself a bootable USB stick that will be able to act as your Recovery Partition in case it gets lost in the drive re-partitioning too.

The boot drive is failing, or you have some other hardware fault. Make a "Genius" appointment at an Apple Store to have the machine tested.

Bad news: without a backup, some data is already gone. At best it is nothing that mattered to you. At worst, you may not be able to recover anything.

Good news: as long as you updated your mid 2010 Macbook Pro with the EFI Firmware Update that was released about two years ago, you can use OS X Internet Recovery to download and install Mountain Lion. You need only a serviceable hard disk and the Apple ID used to purchase ML. Creating a standalone installer is not required.

Computers that can be upgraded to use OS X Internet Recovery

As far as the ability to recover your data though, you must find some way to mount the failed volume first. Data recovery utilities are available, but success is not guaranteed.

Is there a way to determine which components have failed other than what I have already done?

Not with the means at your disposal. The likely points of failure (but not the only possible ones) are the drive, the SATA ribbon cable, and the logic board.

It is not possible to be completely assured the problem is isolated to the HD alone. It is the most obvious and common cause of what you describe, but a communication failure between the drive and logic board is possible. Your Apple Hardware Test results are encouraging, but only negative results from AHT can be considered totally reliable.

It is possible to replace the HD yourself at a cost considerably less than $300; considerably less than $100 if you so choose. Purchase a replacement internal HD and tools (if required) from OWC. The site also has instructional videos so you can determine in advance if you want to perform the replacement yourself.

OWC: http://eshop.macsales.com/shop/hard-drives/2.5-Notebook/SATA

Look for one of the "DIY Kits".

If you ultimately determine the logic board is at fault, or some other cause uneconomical to repair, you can always use the HD you purchased as a backup for your other Macs, which they must have.

A SSD would certainly be desirable, but if money is a factor I would avoid it for now. Your 250 GB is so small that I recommend at least 320 GB. It costs about the same as a tank of gas.

If that fixes the problem upgrade to the SSD when you can, and use the HD as its backup.

When replacing a HD you should always purchase one with more capacity. A 250 GB SSD is roughly $250 and will convey no additional storage, only speed. When you eventually purchase an SSD get one twice as large as what you think you need.