Check for an amanda user or an amandabackup user. If it's home directory is, say, /usr/local/etc/amanda, then look to see if there is an amanda.conf in /usr/local/etc/amanda/daily or something like that. Do an `su - amanda[backup]` and then `amcheck daily`. You'll either need to know the password or do the `su` from root. That will tell you the status of the last run. Also, as user amanda[backup], do `crontab -l` and see if it exists if there is an entry for amcheck and/or amdump.
Those checks will give you an idea of whether Amanda has been configured and/or run. Then you may want to hurry on over to http://wiki.zmanda.com
and brush up on it.
I'm less familiar with ZRM for MySQL, but your approach should be similar. If it is running, there should be a crontab entry somewhere (see if there is a user for it), and it should be doing mysqldumps to disk. Those would then get picked up by Amanda; and, if you have a tape drive or tape library, might be putting things there. The amanda.conf would tell that.