What all are the recoverable points of failure in an Oracle database?
Not really sure what you are asking for here.. If your database is backed up correctly, then there is no unrecoverable points of failure. However most "failures" will result in your session being terminated or at the least being thrown an error. In most cases you wont want to deal with specific errors, but rather simply rollback the transaction ( if possible ) and log the issue for inspection.

If you are testing a 3rd party application, and assuming your are on a standalone database, I'd just test it for session termination to see if it automatically reconnects itself, correctly cleans up the work it was doing, etc. You could also lockup its session I guess and see how it handles timeouts.

Sorry for the vague response, but the question is quite vague.