OK !

First take a look at this picture :

 

 

Basically, what I want to know is, how can I determine the minimal mirror's length so that the observer (blue smiley) can see the whole reflected image (reversed moon)?

you have two of those lines sloping the wrong way.

 

the lines on the side with just the moon should be diverging as you get further from the mirror, not converging.

 

Basically, what I want to know is, how can I determine the minimal mirror's length so that the observer (blue smiley) can see the whole reflected image (reversed moon)?

 

 

You have to look at the angle it subtends, how far away it is and how big it appears to be (2 of those determine the third). (as an example, you should be able to show that to observe yourself, at length L, the mirror needs to be at least L/2, because of the location of the image)