Difference between full and level 0 rman backups

No, you still need archived redo logs. An RMAN backup is not the same as a cold backup; when you restore it you need to apply all the redo generated from the time you started the backup until the time the backup completed in order to get the SCN consistent across all the datafiles and the controlfile.

Let us consider the old fashioned hot backup. In this mode, entire changed database blocks are written to the redo log stream, and the file is copied in the OS. This causes fractured blocks, i.e. the block has changed during the read by the backup. So to recover the database, we restore the DBFs, then the recovery process overlays the changed blocks from the archived redo logs onto the fractured blocks in the file, and we get back to a consistent database.

In an RMAN backup the problem of fractured blocks is eliminated because the blocks are read by Oracle via the SGA not by a Unix tool like cp. However the DBF changes between the time the first block is read and the last block, so it is the same problem, just at a larger scale. That is to say, the SCN changes between the first and last blocks in a backup. The archived redo logs cover this as well.

RMAN gives you the manageability of the catalog, and a means to do hot backups without needing to generate redo at a much higher rate (which puts its own stress on the disks, the backup system etc). However BACKUP DATABASE will not give you a consistent database in and of itself; the only way to do that in Oracle is the cold backup.

Yes, if you want to prevent a level 1 from being created without a corresponding level 0 being available, then you need to do a crosscheck in the script so RMAN knows that the level 0 is not available.

Prior to 10g (or with compatibility < 10.0) Oracle would do a level 0 when a level 1 was done without a level 0. Since you are on 11g, I would expect your compatibility to be >=10.0, so RMAN "copies all blocks that have been changed since the file was created". This behavior seems to have changed between versions and there is some confusion as to what the current behavior is, but your findings are consistent with the documentation.