A drive that will not tell Disk Utility BOTH its Make&Model AND a reasonable, non-Zero Size/Capacity is Dead. It cannot be repaired in its current condition.
I suggest you buy BOTH a replacement drive and an external enclosure to park it in while you re-Install Mac OS X on it. Using an external enclosure postpones issue of Installation from issues of Surgery until later, and allows you to more clearly determine and fix the problem.
If you were forced into Internet Recovery, you can only re-Install the original, "shipped-in-the-box" version of Mac OS X initially from there. Once you have the full Mac OS X running, the Mac App Store will allow you to upgrade to any Previously-"Purchased" version, or choose the currently available version (ElCapitan, at this writing). But getting only Internet recovery instead of Recovery_HD Recovery is a further indication that your internal drive is DEAD.