I am not sure if the bulb would light up or not.
Since GE likes to keep tech info hush hush I do not know the resistance of the solenoid's winding.

If it is relatively high then enough current will not flow through it to light the bulb.

I would suggest you purchase a meter. You can get a decent digital multimeter for under $20.00. You do not need fancy though it is nice if the leads are a couple feet long.
If it saves ordering one unnecessary part it has paid for itself and you end up owning a useful tool.
Most places will not let you return electrical parts so if you order it, you own it.
A couple things to watch when measuring ohms and continuity
1. Always remove power from the machine otherwise you could blow your meter.
2. Always disconnect at least one side of any device you are checking. This eliminates the possibility of measuring an alternate/parallel circuit path.
3. When checking for closed contacts and continuity use the lowest scale (Usually 200 ohms). Then try higher scales. This scale is 0 to 200 ohms so if the device you are measuring is 300 ohms this scale would show an open circuit which it is not, you are just measuring outside the scale's dynamic range.

There is a good STICKY at the start of this forum about it's use.

So I purchased the multimeter today. It appears that there is no resistance at 200 ohms. The next highest setting is 2k. Should I test there?
Yes, When looking for continuity of a part I always try a couple scales higher that where I expect the reading to fall, just to make sure it is a true open.

Also, in order to manually determine if the solenoid is the problem, will I need to hold the arm down for ten seconds? If so, can I do this when the dishwasher tub is full, but there is no power?
With water in the tub, door not latched, put the timer to somewhere in the cycle where the pump will run. Hold the solenoid down then close the door. The ten seconds really does not matter as once the water hits the flapper valve it should hold it in the drain position.

If holding the arm down does not cause a drain does that indicate that the solenoid is not the problem?
Yes or you may have 2 problems.

I am simply trying to hone in on the problem, so I don't spend $40 on a solenoid and then have to buy a new dishwashwer after this does not solve the problem.
I completely understand. I do not replace parts unless I know they are blown or the odds are high that they are blown. If I want to throw money away there are better things than then appliance parts.

If you recall I indicated that I held the arm down during a cycle at one point, and water started to leak to the floor. When I was doing my initial, ignorant diagnosis, I pulled off the wash arm and prodded down that hold with a screw driver. Now it appears that the rubber piece at the bottom is off to the side, which I believe based on further review is where it belongs. But, is it possible I disturbed something which would cause the leak on the floor?
Sorry do not understand the above. I would try to get it back to where it was. Also could be it will not make a difference

Last thing, if I choose to make the repair, will I be able to make the replacement just by accessing through the kickplate, without pulling the whole dishwasher?
Yes.

Would that timer position be just before the rinse and just before the dry, it's not marked but that seems where it would be.
I would put it mid wash. You are faking a drain so it will drain even if it is in a wash cycle.

Also, when I close the door, do I latch it at that time?
Yes. You have to get the motor running.

And to be sure, activating the solenoid is pulling the arm down correct?
Yes. Up is wash, down is drain.

Finally, power is on to the machine?
Yes. Just watch that you do not touch any wires.

It looked like it came from the arm lever.
If you mean the flapper arm lever seal then it is replaceable. Get both the seal and the push on nut (Items 415 and 417 in Section 3 of the parts)

When this initially happended (the water in the tub) I took off the sump cover and the sump cap and pulled the sump inlet aside (pulled the plastic aside) which allowed water to escape,
Sorry I do not understand the above.
Use a small mirror and a flashlight to find the leak, if you are unsure where it is coming from.

So the leak is coming from the drain valve shaft seal. I replaced the seal, and it is still dripping ever so slightly. I haven't put the push nut back on, nor reinstalled the new solenoid. Thoughts?