Hard Drive not showing up in Disk Utility
I think it's time to start thinking about data recovery at this point. Half-performing Time-Machine can mess things up pretty badly (you don't need to be being told this, you're living it).

This is most easily done (read: Unless you want to open it, which isn't terrible, but definitely not for a novice) by holding the "T" button on the keyboard while powering up. This basically is like opening up your Mac, removing the internal drive, placing it in an enclosure, and then connecting it to another Mac. But it does all of that without any disassembly.

Before entering Target Disk Mode, you’ll need a few things:


Two Macs: Target Disk Mode works with Macs, so you’ll need two Macs for this. Each Mac needs either a Thunderbolt port or a Firewire port. [You might need to borrow a friends' if you don't have one]
A Firewire or Thunderbolt Cable: You’ll need either a Firewire cable
or a Thunderbolt cable for this. You can’t do this via a USB cable.
If one Mac has a Thunderbolt port and the other Mac has a Firewire
port, you’ll need a Thunderbolt-to-Firewire adapter cable.
FileVault Disabled: Macs now enable FileVault encryption by default,
which will prevent you from accessing FileVault-encrypted home
directories over Target Disk Mode. Before using Target Disk Mode, you
can boot the Mac normally, open the System Preferences window, select
Security &amp; Privacy, select FileVault, and temporarily turn it off.
You can turn it on again after using Target Disk Mode. [You can't do this step, so we'll just hope it was off...]
No Firmware Password: If you’ve set a firmware password in the
recovery environment, you’ll need to disable that first. [If this makes no sense, you probably don't have one]


With Target Disk Mode, you can treat a Mac’s internal drive as an external drive and boot from it, just as you’d boot from a typical external drive. This actually lets you boot the OS X system from one Mac on another Mac.

You can also simply plug it into another Mac, and download your important files. Once this is done, we will want to reinstall Mac OSX onto the iMac. 

To Re-install OSX once your files are safe and sound:

Immediately after hearing the Mac boot chime, hold down Command+Option+R – if you see the Apple logo you waited too long and need to reboot and try again*
OPTIONAL: You may or may not see an option to join a wi-fi network, this depends on whether the Mac can access any saved networks from OS X or not When you see a spinning globe icon, Internet Recovery mode has been entered with a message saying it can take a while, a progress bar appears as the recovery functions are downloaded Starting OS X Internet Recovery Mode on a Mac to reinstall system software.

When finished downloading, you’ll see the familiar “OS X Utilities” screen, choose “Reinstall OS X” to begin the re-installation process of the Mac operating system Install OS X from Internet Recovery.

Select the destination and complete the re-installation (or installation) of OS X as usual.

Sources [First Half, Second Half].