So that means you can only test read-only features of your site then? I can't imagine you want dummy test data floating around your production servers.
I don't know if this really qualifies as an acceptance test, it's more like an advanced version of Pingdom.
It really depends on how your application is structured. If only the test sees the test data, you can easily do it on the live site. If there were a hundred test accounts in gmail sending mail back an forth to and from other services, you would never know.
Yes. I have cron-jobs running tests on live sites looking for broken things.