With a working alternator your battery voltage should be from 13.0 volts up to 14.9 volts. From your readings I will say the alternator is not charging the battery. Your battery voltage should be 12 volts. Get the battery charged up first. A lot of shops have a newer type of charger that will charge the jell type AGM, Optima Batteries. I would also check your system for a draw. I would disconnect that new head unit you installed. Seems like the problems started with that install.
I had the same idea as you just a few minutes ago about disconnecting things. The battery read 13.30V this time with the jumper cables.
Before that though, I had the multimeter propped up still reading the battery while I turned ACC on and started disconnecting things. I disconnect power and ground to the head unit and wideband gauge. When I went from having no power applied, to power applied, the voltage would drop by around .08V and stick there temporarily. I then proceeded to disconnect power and ground to the head unit and wideband individually. I noticed when I had power connected to wideband, and connected the ground, voltage would drop by around .10 rapidly, then when I took the ground off, it'd climb back up, connect ground, voltage back down and it sticks at a certain voltage. I did the same with the headunit and it didn't have nearly the same symptoms. So I immediately thought the wideband to be suspect.
So I hooked up jumper cables again, started the car, took off the cables and monitored the voltage with the grounds taken off both the headunit and wideband at the same time, but with their power wires still connected, and the voltage was STILL dropping. I took off power to the headunit and same deal. I however didn't take the power wire off the wideband yet, but I figured that'd be a lost cause.
So right now, I'm at a standstill and more confusion. I'll start tracing more wires when I have more free time. But as off now, I couldn't visibly see any exposed wires.
What is the item that you are cooling the "Head Unit"??? Please use the proper terminology such that appropriate advice can be issued. It sounds as though the Alternator is good but the wiring is faulty and not connecting the alternator to the battery properly. Do you have the wiring diagram for the 240Z? What is the "Digital Gauge"??? What does this gauge indicate? The 240Z has the ammeter in series with the alternator and the battery to measure charge/discharge current. What is the ammeter indicating throughout all these exercises? An appropriately charged battery should be slightly greater than +12 volts. When cranking the engine for starting the voltage should not go lower than 8 volts DC. If it goes lower there are bad cells in the battery. When the alternator is initially charging after engine start, the alternator voltage should be +14.x volts. As the battery attains a charge the alternator voltage should taper off somewhat from +14v yet still stay above +12v. The newer alternators need only a small amount of current from the battery to provide excitation for the alternator field to cause it to generate 12 - 14 vdc. There may be an excitation return (2 wire excitation) to ground or battery negative to complete the circuit. Normally the case of the alternator is the return or ground path for both excitation and heavy current to charge the battery. The other heavy post or connection on the alternator connects back to the battery through a heavy current (low or no resistance to current flow) fused wire. Older alternator charging systems had more (complicated) wiring than newer charging systems.
Hope this helps. Will look at the 240Z wiring diagram to see which OLD alternator system wiring IS NOT NEEDED.
Source: Suddenly not keeping a charge.