Well the brilliant folks at Apple got back to me yesterday after Sunday's crash and support call that I wrote about earlier.
I have had to restore the phone (which didn't necessarily work on the 1st 5s I had) and set it up as a new user, from a new user account on my MBP. For the next 48 hours I basically have a very expensive lightweight paperweight that can make calls. I can not install any apps or download anything on to the phone during this time. I can't access my iCloud to establish my contacts or any of that jazz. They want to see if it crashes with just the original software (obviously it's still running 7.0.2, though) or if other apps are interfering with the system.
I've run about a bazillion pages of safari open along with mail, camera, photos, iTunes Radio, text, well basically most of the apps that come on the phone (aside from iTunes, Game Center, newstand, passbook, and App Store since those are pretty pointless without my Apple ID) and so far, no problems, but I wouldn't expect there to be with so few apps taxing the phone.
Tomorrow, they'll be calling back and we'll go through the process of troubleshooting my apps to see which of them is screwing up the system. I guess the engineers will have a sequence of apps they want to try. Right now I'm very frustrated with having a very expensive phone that I can't do much more with than a basic flip phone. If it can't handle apps, then what's the point, right?
And viewing websites that have dedicated iPhone apps available is painful in Safari. Half of the functionality of the sites disappears. Facebook is particularly unusable, at least for me. Text disappears as I type only to reappear two lines later, pictures can't be viewed properly, it's a mess.
I'll keep you guys posted as I soldier on with the Apple engineers.