> How do I know without looking whether the fake store method REALLY stores the doc, or whether someone set it up to just return OK?

Why would you not look into the fake implementations? If it's just a two liner that takes the parameters and writes them to disk, then that's a LGTM from me. You would not test every getter/setter method in any of your classes, would you?

If it's more complex than this, you can absolutely write tests for your fakes. In many cases fake data-stores use tested backends, and tested client libraries. If the client wrapper is complex enough then you can/should test it as well.