Electronic – Why does resistance reduce when a conductive fabric is stretched

The wire gauge you need is a function of several things.

• The acceptable voltage drop or power loss (that appears to be the only thing considered in the website you linked). The voltage drop (and power loss) is proportional to wire length and inversely proportional to the cross-sectional area of the wire- in other words inversely proportional to the square of the wire diameter (assuming constant current).
• The acceptable temperature rise. This is a function of the number of current-carrying wires bundled together, the environment (maximum ambient temperature and air pressure or altitude, for example), the insulation type, the wire type (some types of wire are plated to withstand higher temperatures than bare copper without corroding).
• Regulatory requirements and other considerations- for example, the wire may be rated for 200°C insulation, but you might not want the wire to run that hot.
• Fusing- the fuse or circuit breaker should protect the wire in the case of faults such as overload or short circuit.

Very short lengths of wire can depend on heat sinking through the ends (indeed, in a vacuum, that may be the main heat loss mechanism), but usually that's not taken into account.

Normally you'd run through a checklist such as the above to make sure ALL the requirements are satisfied simultaneously, so you might find that using PTFE insulated wire allows you to use AWG 18 wire, but because of the voltage drop limitation you'll have to use AWG 12 wire.

After you run this test for 10 minutes, wipe off the water and take a close look at the copper where the water drop was. You will see some discoleration. Essentially the copper was corroded a bit where the water drop was. As you probably notice, bare copper left in the elements is no longer that bright copper color after a while. The same thing happened to your copper electrodes, except that the electric current speeded up the process.

The reason the resistance goes up is because the corrosion layer has significantly more resistivity than copper. This is one reason the mating surfaces of electrical connectors aren't made from copper. They are usually made from material that doesn't oxidize, like gold or nickel, or forms a conductive oxide, like tin.