What processes run while my Mac is asleep?
Nothing will run while the computer is asleep. Unlike humans, where sleep is different than death, when a computer sleeps it is essentially equal to being powered off. 

There are a few subtle differences, the primary being that it wakes up faster than from a cold boot. This is because there is enough power supplied to the RAM (memory) to keep it active, but this does not mean there is power supplied to the CPU (heart), or that any processes are active.

Any processes that are active are below the operating system, in the BIOS of the computer. This is where the "Wake On Lan" feature resides, for example.