First of all my compliments that you are trying to do this so thoroughly ! That kind of stamina will get you very far, but you should avoid losing sleep over the subject, because then it becomes counter-productive.
To answer your items:
1. I was referring to the voltage over the piece of fuse wire. As I understand it, you vary a series resistor and measure the voltage over and the current through the fuse wire. So yes, the battery voltage is relevant and the resistance of the variable resistor is relevant as well. For the fuse wire it doesn't matter which one is varied.
2. As you hopefully distilled from the other threads, the current can heat up the wire and thus increase its resistance somewhat (until the wire melts and thus fulfils its useful function of protecting against too high a current). Pity the experts started making things more difficult than necessary.
3. You simply reverse the connections at the power supply (the battery). Depending on the equipment, switching the connections on the meters is also necessary (but some meters actually put a minus sign in front).
4. The whole experiment demonstrates that current through a resistance is dependent on voltage: different voltages, different currents. The relationship appears to be a straight line through the origin (right ?), meaning that the relationship can be mathematically described as I = constant * V . I use that way to write it because of your
post #3 in the other thread. The other, completely equivalent way to write is of course more the familiar V = R * I Ohm's law.
Good thing you ask about the distinction between voltage and current. Current is indeed rate of flow of charge. Now what's the other one ?
In the analogy I remember form long ago, we compared current to rate of flow of water through a hose or a pipe and voltage to the pressure that pushes the water through. Voltage is tension: how hard the charge is being pushed. Generally voltage is a cause and current a consequence. Depending on the possibilities to 'ease the tension'. If there is no path, there is no current. A wide path offers low resistance, a narrow path offers more. Long path: more resistance, etc. The analogy only gets you so and so far, but by then you're hopefully familiar with the phenomena.
Good luck !