When to use analog PID
Since I've essentially created an analog PID controller with only
  proportional coefficient in the second schematic - do I need to
  implement a full-scale analog PID controller (like the one in this
  question) with an integrator and differentiator or will the second
  schematic be sufficient?


If you don't need to adjust the I or D component, then don't include them. Many controllers are only PI controllers. The question is, "what do you need to be adjustable?", and "what controller is suitable for the final design?". No two components are created equal, if component tolerances affect the operation of the controller (which they almost always do) then you'll most likely need to adjust the gain. If the pole is changing with component tolerances then it might be wise to include I to compensate for changes in frequency. 

The best thing to do would be to include the full PID controller for testing, and then reduce it to a bare minimum for production.


  If I only need to lower the gain - how should it be selected?


Either by trial and error, preferably with a variable resistor or by actually calculating what the gain should be with a controller/plant model and by using control theory. 


  When exactly (excluding cases with mechanical systems and such) do I
  need to implement a PID controller or lower the error amplification
  gain? For example, a simple opamp voltage follower is also essentially
  a controller that tries to set it's output voltage equal to the input
  one and somehow it is fine without PID or gain reducement.


The feedback path of a simple voltage follower has a gain of 1, and the plant has a gain of 1 which makes it a stable system. The problem with control systems is they go unstable when the phase on any given frequency reaches -180, then you get positive feedback.