Let me see if I can summarize the few tips to mitigate issues, salted amongst the speculations and complaints in this thread. These seemed to have helped me reduce the quantity of Air problems:
In settings, under Accessibility, turn on Reduce Motion
Turn off Background App Refresh
For Safari, remove all bookmarks from the bookmarks bar
Under icloud, turn off Safari
I also have keychain set to off
Other things I have done to reduce system churn:
Under iTunes and App Store, automatic updates to off
Location Services off for all apps except those that really need it, like maps
Manual fetch of mail
Then there are the more obvious, as with iso6:
Close running apps when you are done with them
Close Safari tabs when you are done with them
I still have crashes, but not nearly as often.
As I commented earlier ios7 is not matured. Memory management is not as good as ios6. Safari is flaky. Some apps are not behaving well under 7. There are interface issues not yet resolved. Etc etc as with complex, immature software. It may be that the 64 bit version of ios7 is more problematical than the 32 bit version at this time.
Is lack of memory a fatal flaw? Yes I was disappointed the Air didn't come with 2gb of ram or more. But my expectation is that the software will improve so that the things I use the ipad for will work as well and better than with my old ipad4. I like my Air and am not sorry I bought it. For one thing, I have noticed that Candy Crush, a graphics intensive app, drains the battery significantly less on the Air. Just an example.
For "power users" maybe the Air will not be fully satisfactory. But then, the Air is a Tablet. It's not designed to do things a laptop can do. Lots of options these days for those who want a box that tries to bridge that divide. There will always be websites that are not fully compatible with iPads, but many will make the effort to improve because the audience is huge and growing.
I will do the hard refresh to see if things improve. If not, I am in no hurry to exercise my warranty for a replacement. I'm happy enough and will await further developments.