As douglasarcidino mentioned, ESX 5.5 and 6 should work on your DL380 G6 with varying levels of support. As you've mentioned you are using the HP image, tried 5.5, and updated the firmware, all of which I would have tried too.
There's always a chance that it's hardware related. Try running HP Insight Diagnostics or even MemTest86+ to see if there are any issues. The recommendation to try booting from USB would isolate any potential HDD issues as well.