There are several things that can cause the mysql spike, from a stock sever (not optimized) to traffic spikes, to corrupt DB, etc.
Log-wise, I don't see anything suspicious.
Relevant Skills and Experience
I run a web hosting service and have extensive server troubleshooting experience.
Proposed Milestones
$100 USD - Issue located and fix implemented.
PS: The one hint I see are those CPUs at 1.3GHZ, which means they're junk, but I will try optimizing mysql first, after having a good look around to get a feel of the system