Any way to make the mirrored mesh more idiomatic? If you turn on the edit cage for the mirrored side, then if you try to move the mirrored side vertices, they actually behave as if you were trying to move the non-mirrored side, i.e. (assuming you're mirroring the x-axis) you have to move the mouse left if you want to move the vertex right and right if you want to move it left.

VERY confusing and super annoying. Is there an  addon or something that changes this behaviour so that you can move the mirrored vertices as if they were actual vertices?
No, there is currently no way to do this. You only select the vertices that you see on the geometry generated with the Mirror modifier, but what you transform are the original vertices. 

You could temporarily mirror the whole object by multiplying its scale by -1 on the mirroring axis so the original geometry is on your preferred side. You can just enter -1 after pressing s followed by the axis, in this case x in object mode, or you could also use the dedicated Mirror operator(Ctrl+m) to do this, or just enter *-1 in the numeric input field in the n panel, Transform tab, under Scale or in the Properties Editor's Object tab.