I'm really glad my post brought up so many thoughts and so much speculation. Thanks, everyone, for your thoughts, and please keep them coming. Here's another idea I didn't get to include in my original post:
What if the Aes Sedai symbol includes an explanation for the Shadow? The AS symbol reflects the Wheel as whole, and it also reflects the relationship between saidin and saidar. [I thought it represented the DO & Creator as well, but that's not important now] Is it possible that Shaitan IS the part of the symbol that is neither saidin nor saidar?

By this I mean that the DO is the lining around both halves of the symbol; he is also, probably, the opposite-spots that appear in each half of light and darkness. This would explain why Shaitan can be accessed by both men and women - he touches both parts of the OP, so they can both be touched by him.

I'm not sure how that info might lead to his defeat, but I've obviously been looking at ways to analyze the Shaitan's place in all this.