Common collector :current amplifier or voltage follower?
It is known that this configuration amplifies the current. How can
  this happen when it also provides a ....'fixed' voltage on the
  emitter?


Because..... it doesn't quite provide a fixed (or constant) voltage on the emitter.

As the load current increases....


The emitter voltage drops slightly and
That slightly increases the base-emitter voltage and 
That, in turn causes more current to flow into the base and 
That causes more current to flow in the emitter load and, 
That has the effect of regulating the emitter at a roughly (not perfect) constant value. 


It's a reasonable voltage regulator but not perfect. It relies on negative feedback also.

The "system" is driven by slight changes in \$V_{BE}\$. Those slight changes can cause quite significant changes in base current.


  Question #2 Why does this configuration has high input impedance? I
  imagine because the current and voltage needed to drive the base are
  very small and equivalent to a very large resistor. Is this correct?


And


  sorry. bjt base input. that should be high impedance – sergiu
  reznicencu


If the BJT can avoid saturation by ensuring that the collector voltage is a few volts higher than the emitter voltage then, the current into the base is largely determined by emitter load current ÷ hFE. The impact of this is that the impedance looking into the base is approximately hFE x emitter resistance. "High impedance" doesn't really mean anything but it's certainly hundreds of times higher in impedance than the emitter load resistor.