As Alain mentioned, adding 'hollow' to the right-side plane will cause the
reflected spotlight to show up nicely in the media.

From playing with your code, the photon block in your 'target light source' is
apparently not needed; the result looks the same with or without it.

> The very final step was to enlarge the mirror even further, when the
> reflected spot disappears completely (see image f).
>
> So to summarize I would like to understand:
> 2) Why is the transverse profile of reflected spot different from the
> direct one, and how to recover the initial profile (this is the main
> issue I have).
> 3) Why the size of the mirror matters and why the light disappears for
> large enough mirror...

These two effects seem to be interrelated... AND related to the photons 'count'
in your global_settings block. Given a mirror size of 1, the reflected
light-spot appearing on the left plane gets sharper with a higher count--
10000000, for example. Strangely, the count also affects the result when the
mirror diameter is increased; in other words, the relative blurriness of the
light spot is an interplay between photon count and mirror size. Which is rather
odd!

It's *as if* the given count of photons is being applied to (or spread out over)
the full area of the mirror surface (not just in the area where the spotlight
cone hits the mirror.) By increasing the mirror size, there are apparently less
photons 'for use' in the unchanged spotlight cone-size...requiring a higher
'count'.

This is my guess, anyway!

Here is a simplified version of your code, to test this. Vary the photon count
vs. mirror size.

----------
#version 3.8; // or 3.7
#declare MirrorSiz = 1;   // [1 | 5 | 6]

global_settings {
   assumed_gamma 1
   ambient_light rgb 1
   max_trace_level 10
   photons {
      count 10000000 // 100000  original
      media 100, 2
   }
}

background{rgb .1*<1,.5,.5>}

camera {
   location <0, 0, -10>
   look_at  <0, 0,   0>
}

light_source { // ambient light
   <0, 10, 0>
   color rgb 1
   photons {
      reflection off
      reflection off
   }
   media_interaction off
}

plane { // left side plane
   x, -5
   pigment { color rgb 0.7 }
// photons { collect on } // [apparently not needed]
}

light_source { // target light source
   <0, 0, 0>
   color rgb <0, 1, 0>
   spotlight
   radius  10
   falloff 10
   point_at <1, 0, 0>
   looks_like {
      sphere {
         <0, 0, 0>,
         0.1
         pigment { color rgb 1 }
         finish  { ambient 1   }
      }
   }
}

   cylinder { // mirror
      0, 0.5*z, MirrorSiz
      texture {
         pigment {color rgb 1}
         finish {
            reflection 1.0
            ambient    0.0
            diffuse    0.0
           // specular   1.0 // [apparently not needed]
         }
      }
      photons {
         target
         reflection on
      }
      rotate 90*y
      translate <2, 0, +0.0>
   }

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