To go back to the original question for a moment, I think that, as has been said, the duties vary for the different queens. However, a few things seem to make sense:
-The Knight acts as a bodyguard for the Lady when she needs one (possibly whenever she ventures into the mortal world). We didn't see Slate with Maeve in PG because he was... not free to leave. But after SK, we NEVER saw Lily without Fix (though we did once see the inverse). Since the Lady is about as strong as a really powerful wizard, it's possible that the power of the Knight is placed with her more than with either of the other Queens (based on what we've seen and what Jim has said, NEITHER woman needs much in the way of bodyguards, even away from their centers of power).
-The Knight can be sent on mundane tasks. Generally they won't be wasted on things like coffee fetching (unless it's a "If you're going there, can you get me one?"), but retrieval of items seems to be on the list.
-It's not outright stated, but thus far only the Queen seems to be able to ORDER a kill. In SK, Maeve asked Slate to
presumable for a ritual. But why not kill her outright? Just a theory, but maybe that's a Queen-only duty.
-Not sure what s/he does for the Mothers. And honestly don't want to dwell on it too much...