Wiring – Splitting one main low voltage power line into two

Yes! For goodness sake, do NOT put any commonly available potential (ie 120V) on the lines, they make very easy to use tools for this. Most sources in your house will be voltage sources, vice current sources, so how much current results is a factor of the resistance in the line. Because these speaker wires will have low resistance when you find both ends, you will get a LARGE current in the lines which could destroy your fuse/breaker or even cause a fire.

What you are describing/seeking, though, is essentially a resistance measurement (ohm reading). Get a multi-meter, the "ohm"/resistance measurement ("upper-case" omega) sends a small current down the wire and measures the voltage, to determine the resistance (using the equation v=ir). In your case, you know you will have found the other end when the measurement of resistance goes to near zero (after starting at off scale, or pegged). I would wrap one of the speaker ends around the black lead (color chosen for no particular reason) then go to each wire on the other end until you get the near 0 reading of resistance.

If the wires are too far apart, make an extender out of an available piece of speaker wire. Simply twist the free piece of wire to one end of the speaker wire on the wall to extend it far enough that your red and black meter leads are long enough to reach both ends. Then continue with the resistance measuring technique described above. Now, use the red lead of the multi-meter to go through each wire until you find the right one. Mark those wires and move on to the next.

That looks like a standard appliance cord.

Look closely at the cord and you will notice one conductor is ribbed. Follow it... It should lead to the neutral side of the plug and the white wire on the dishwasher. The ribbed side being neutral is an industry standard, if the plug is molded on to the cord it will follow this convention.

You are correct about the wide BLADE being neutral, but wrong about how to follow it.

I think I can see the rib in both pictures, when enlarged, and it appears correctly wired, but you should double check this.

Edit addition:

There are different type of rib on SPT cord, the picture shows a single rib. There are also versions with multiple small ribs, which is more obvious.

On the straight cord picture in the question the ribbed conductor appears to be the one without the writing.