I ll take your several questions one at a time:
Bigtable is the system on which AppEngine s datastore is built. It is effectively a distributed hashtable.
Authentication is required in that you must have a Google Account; you must have signed up for AppEngine; you must have created an application within AppEngine. You application will be able to access the datastore, and if you are logged in to your application s Admin Console, you can use the Datastore Viewer to inspect the contents of your application s datastore.
The data will be stored on Google Servers.
There are no tables, per se, but you can use the Datastore Viewer to view entities that reside in your application s Datastore.
No, you can not ever view the Datastore s contents that were created by other applications. Each application s view of the Datastore is completely siloed and has no connection to that of other AppEngine applications.