Here are some commands you can run as queries that might help:
select databasepropertyex('yourdatabasename','recovery') tells you what recovery model the database is using,
select databasepropertyex('yourdatabasename','isautoshrink')
tells you if AutoShrink is on, '1' is yes. I can't find a command that will tell you if backups are being taken, or when the last backup was, sorry, maybe someone else here will know.
I'm also fairly inexperienced at doing the backup thing, but my log database is the bit that grows fast and setting the recovery to simple with autoshrink drops the log file to a reasonable size, I am aware though that this means that I can only restore to the last backup and therefore any changes I make in between backups may be lost, hence I backup to disk every night and transfer to another machine when I remember.