Nah don't get sidetracked by the EZK until you figure the alternator problem out (if your supply voltages are wrong, your computers are the electronic equivalent of drunk). The alternator is a simple machine, it needs a circuit with the battery and it needs the exciter to supply the 12V field voltage required to kick it into gear.
If your battery warning lamp is on when the engine is stopped, and goes out when the engine is running, that suggests the alternator is at least somewhat functional. You may have parasitic losses from wiring problems.
It looks like you are testing voltages across the battery terminals. You should test the voltages using the terminals on the back of the alternator, and use the alternator chassis itself as your ground reference. If you do that, and the numbers look normal, you've ruled out the alternator itself and most likely have a wiring issue.
I have some spare voltage regulators if you want to try another one.
