Your diagram is confusing as many of your arrows are pointing in the wrong direction. However, the answer to what I think you're asking is yes, you have to pass the audio signal through the Mini Effect Gizmo, or it won't have any effect on your signal. I'd think you would probably want the Mini Effect Gizmo in between the guitar and the input of the Mark V, as the pedals you've got connected to it in the diagram would sound best in front of the amp... then you'd be using the Axe-FX for the signal processing in your effects loop.
The MIDI signal is only a control signal - it sends data from the foot controller to the Axe FX, Mini Effect Gizmo and Mini Amp Gizmo telling them which functions / loops / presets to choose. A couple of things I've noticed from the diagram - you'd want to connect to the MIDI Thru, not the MIDI Out, of the Axe-FX... and you probably won't need the Midi Solutions Thru box unless you've already got it - simply connect the MIDI signal from foot-controller > Axe-FX > Mini Effect Gizmo > Mini Amp Gizmo using the MIDI Thru each time. You can safely get away with chaining four devices without needing a separate Thru box - it's only if you were to want to add another MIDI device or two that you start to get data signal dropping which can cause glitching in the devices.
