First, a correction. I assumed that the lug which the cable from the alternator attached to was the fusible link. It is not. The fusible link is the white box in the center of your first picture. The lugs the black/red cables attach to appears to just be a connector.
Here is a link to the wiring diagram (on page sc-20):
http://www.nicoclub.com/FSM/QX4/2001%20 ... System.pdf
Not sure whose pink fingernail that is in the last picture, but I will assume it is your girlfriends...
As to your questions:
There is one more ground point on the negative cable down near the bottom of the fender just before the cable turns upward to go to the engine block - at least it is that way on my 02.
The fuse block looks ok - I think you caught things in time. Some melting, but not bad. I would unbolt the large black / red cable on the underside of the fuse block, clean it up, and retighten it. As a test, I would set the voltmeter to ohms, and check the resistance from the top lug of that connector to the positive battery cable terminal. It should be zero ohms. Probably will be, but its an easy check.
Then check resistance from the top lug of the connector to the other end of the short black / red cable where it connects back up to the bottom of the fuse box. Should be zero ohms as well.
Clean up your ground points on your negative battery cable, and retighten. After you have cleaned the connections for the large black / red cables, zip everything back together. You may get lucky and find that the spot where it was overheating just needed to be cleaned up and retightened.
Can I guarantee it's the alternator?. No, you would need to check all the circuits going to / from the alternator in the wiring diagram. There could be a bad connection / corrosion, etc. causing a problem. The alternator has to get correct voltages from the smaller circuits for the regulator to work properly.
If the connection still gets hot after this, I do think I would let somebody check alternator output and see if they can figure out what is causing the problem.