Ridiculously slow MacBook Pro
Hard drive cable failure is almost guaranteed to be your problem. We have a deployment of around 70 2012 13" MBPs for staff members, of which around 10% have had a hard drive cable failure. I had a suspicion you were going to say you had the same model from the symptoms you described.

The symptoms on our problem MacBooks are exactly the same as yours with the MacBooks running excruciatingly slowly on the most basic of operations. Replacing the cable has restored full speed without any further work needing to be done. Simply re-seating the cable's connector on the logic board has worked on a couple of occasions - in the short-term - but these have generally slowed down again, sometimes completely failing.

The odd thing is some of these MacBooks haven't been subject to any abuse and the cable has showed no signs of damage. However, the simple fact is that replacing the cable has worked.

The cables are available online (genuine cables can be had at a reasonable price on a certain auction site) and can be changed in under 10 minutes. The fiddliest part is disconnecting and reconnecting the IR receiver/status LED as there's a tiny latched connector to undo, but otherwise it's an easy fix.