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This has to be fixed, whether it's related to your problems or not.
For reference,turn on the parking lights and measure the voltage from one side of the fuse and the clip that's holding it. Then measure the other side. Just like you did the ground connection.
Now remove the fuse and clean the clips that hold it. An emeryboard for filing your nails works great. Since your g/f insisted, use one of her's. Reinstall the fuse and measure the voltage drop across each side again. It should be less and the fuse should not get hot.
That's one half of the fuse problem, now you have to figure out why it needs a 15A fuse. A wire diagram will help you locate everything on that circuit. Check to make sure you have the right sized tail lights, the tail light grounds are solid, the wires that lead to the back are sound.
In a 1200, the dirty fuse thing usually shows up as a melted fuse block at the headlight fuse. These intermittent electrical problems are very difficult to track down. I once let my car idle in the driveway for an hour while I disconnected and connected everything. Seats, floor mats, glove box, everything came out. The neighbors must've thought I was on crack for sure. I eventually found the MSD tach adapter had shorted out and wiggling the wires would stall the engine.

OK, I sanded down all the connections between the + terminal on the battery and the fusebox. I sanded all the fuse clips. The one fuse still heats up but not to hot to touch.
The white wire coming through the firewall is quite warm as well. All the voltage at the fuse box is 12-13 volts.
What is an interlock solenoid and why does it raise the idle and decrease the apparent voltage when I pull its fuse?
Is the rear window defrost supposed to work, cause mine takes 20 min before I notice the fog breaking on the lines. I tested the voltage on the teminals at the window and got

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Best Logic I've heard in a long time.

Nah man, the electric rear window is just for looks ... you can disconnect it.

What is an interlock? We don't know since Datsun 1200s (B110) did not have them. What I would do? I'd go to the library and check out a B210 repair book and see what the part is officially called, and how it's supposed to be wired up.

It sounds like somehing is damaged and needs to be replaced or disconnected.

Just to put some more info out ther on this problem, I took out each fuse individually testing to see if the power dip was bypassed by eliminating the fused circuit and the problem was not affected. I then assumed it was pre-fusebox so I ran a new wire from the battery to the fuse box to jump what could be the problem and still the dip is there...

any more thoughts?

Your voltage across the battery terminals with the engine running should be 13V9 to 14V3 ideally.
Over 15V as you said you were getting is way too high. If your rotor circuit in the alternator is fed full field current ( causing overcharging)it will overload the engine and cause it to stall. This will also drop ignition voltage. My advice is get the alternator and regulator tested together by an auto-elctrician or mechanic to be certain.

The alternator and regulator are brand new

Somethings still not right with the charging, 15V is way too high.
Check at which point he reg is sensing batt voltage.

the problem is present when the car is not running as well.

I've reread this post and I have no more ideas. Can you please refresh my/our memory as to when the lights dim?
If pushing down on the brakes dims the headlights, have you tried holding the brakes down while you disconect each taillight?