Water report: Your utility will have this info. You should be able to check their website, hopefully a recent report is posted there. Otherwise you can email and ask I suppose.

Getting your water tested: Should be a number of places to do this, from water suppliers to assay labs. I'm sure you can find something on the web easily if you need this done.

So your manual doesn't include info on how to clean out the bits of the machine beyond the the detergent tray? Hmm.. Might need to find service manual info then - probably can find that online too. Removing the spray arms should be pretty straight forward though, as well as any covers over the impeller housing or filter areas. Should all just unscrew/remove by hand without any tools.

If you're going to run additional loads to see if the problem persists, you should really try to get the machine itself as clean as possible of any residues.

The softened water testing is pretty easy to do too, but still make sure the machine is as clean as possible, and maybe don't do this first - just to make sure you test one variable at a time. If you're gong to try this, use Calgon powder, not liquid. I remember that one from the many months I spent reading GardenWeb.