This type of control is not made to reverse. It might be possible, but there is nothing in the rigging book for reversing these.
If you shift into forward, it should pull the cable forward. Likewise, if you shift into reverse, it should shift the motor into reverse.
The throttle should pull forward to advance the throttle on the engine. It is the one in the picture that is the outside one on the shifter.
This also looks like it is a right handed shifter. Which means, if you right hand is on the shift handle, your thumb should naturally go to the trim switch on the handle.
Your question on the Mercury prop...you can use a Mercury prop on the Honda 90's...even those with Honda gear cases. You just have to have the right thrust washer.
I do not know how the shift rod could be incorrectly installed. The Mercury gearcase has a shifter cam the attaches to the vertical shift rod going into the gearcase. I do not know if you can install these 180 degrees from normal operation and cause the gearcase to be "backwards".
I have not had to rebuild any of these...they usually coming in demolished and we just replace them with an updated gearcase. You might check with a Mercury mechanic. I would think Justintime would probably know.
Basically, if you push the shift lever on the engine forward and the engine goes backwards, then something is going on in the gear case.
Mike