Does the DA-42 automatically lean to clean fouled spark plugs? The DA-42 has an automatic run-up method. Specifically, the pilot in command is to hold two buttons (one for each engine) until the run-up is complete. Since the DA-42 has no manual mixture control, does it also automatically lean the mixture when foul spark plugs are detected?
Depends on the variant.  Most DA-42s are powered by turbocharged Diesel engines and do not have spark plugs.  And yes the DA-42 uses an automatic run up cycle in order to test the Engine Control Units (ECU).  The test button is depressed and held while the automatic ECU cycle runs on both ECUs for each engine; the pilot simply monitors the engine gauges for signs of abnormalities.

Now the DA-42L is an exception to this rule as it is powered by Lycoming IO-360 gasoline engines.  It makes use of more conventional run-up procedures.