Just did a full Hardware test... everything passed with flying colors?
That's good... for a start.
Download and run Rember or Memtest to re-test the memory. Both do a better job than the AHT.
Run the RAM tests in loops, 10 or more iterations, to thoroughly stress test the memory. Random errors may not be caught in a single pass test cycle.
Should additional RAM testing prove to be error free, and with the AHT not reporting logic board, VRAM or other errors, one may be able to focus solely on possible software issues .
However, hard drive sector damage or sporadic CPU trouble could still be the cause...
I would for now, focus on the hard drive, and the possibility of corruption within the data for system software- the OS.
Disk Utility provides verify and repair options for the hard drive.
DiskWarrior can do much better.
If you have Tech Tool, try it.
A severly corrupt system can be irreparable, with erase and install as the best option. Even if repaired, sector damage would continue to cause problems.
To rule out possible sector damage (and the data contained within those sectors) as the source of your troubles, a security write with zeros (1 pass) would map out bad sectors, and disallow use of those areas in subsequent software installs.
If the drive is 4-5 years old, simply replacing it would probably be better.