Further update: I realized that even though I had selected to not sync photos, I still had many, many photos on the iPad. I tried deleting them within the Photo app but they couldn't be selected. That meant they were phantom copies. Via iExplorer I found them in "thumbs". I deleted them all and voila, all but 4GB of my Other category were gone. It doesn't explain why it happened, nor why Apple's own guidance directions doesn't fix it, but they were gone. However, I still have 4.3GB left in Other. I think, not coincidentally, that is the same size as my Apps. I tried to find the apps on iExplorer and delete them. No dice.
So, I call AppleCare to take advantage of the $99 I paid for that. I would hate to see what unpaid service is like! (1 hour from the time I fill out a form to: schedule a future call, somebody who is basically just an info-taker gets on and then after each question, puts me on hold for five minutes to research or talk to somebody who actually knows. She tries to get me to do the same steps I already read about on this site.) 1/2 hour into the call they finally transfer me to a "real" Senior Technical Advisor. He tries to tell me that the 4GB is normal - that the iPad needs about 4GB of space to house the software. I say not so fast - the capacity shows 57GB. That means iPad has already taken 7GB of my space for software, PLUS I am supposed to give up another "normal" 4GB to Other, for a total of 11GB for their base install? I don't think so. That would leave a 16B iPad with 5GB of available space. Not logical. The tech tries to find an iPad there to confirm with me, about what I already know better than their tech guys - that the 4GB is already accounted for in the "capacity" summary. If the 4.3GB was "normal" then why did I have the 23GB before? That was obviously not "preinstall" apps. Off he goes for another 15 min to research... nice enough kid, but he is researching what is ultimately not the root issue, wasting my time in the process.
Long story short they couldn't figure out why I couldn't restore the device, why "other" was there, etc. I ultimately just selected "erase all content" on the iPad and wiped out everything. It did solve the problem temporarily but I decided to take it in - it is only one week old after all - and get it replaced, which the Apple store did.