Q:

What was the Question that led Euler to his Investigations on Polyhedra?

The question that led Euler to his investigations on graphs is the well-known question related to the seven bridges of Königsberg, and  that story is a must in every introduction to graph theory.  
My question is whether it is also known what led Euler to investigate on polyhedra; was it a 'recreational' question, or was it a mathematical research question of his contemporaries?
A further question would be whether Euler continued his investigations on polyhedra after having discovered his famous characterisation of polyhedra of genus 0. 

A:

May I suggest this article by Joseph Malkevitch, an AMS Feature Column.
Note particularly his point below (screen snapshot--not searchable)
that polyhedra were simply not viewed,
at the time, in terms of vertices, edges, and faces:

 

This point is also made in Imre Lakotos's
Proofs and Refutations (Cambridge Univ. Press, 1976). 

Addendum. Joe Malkevitch sent me this relevant quote in a letter from Euler to Goldbach: "Recently it occurred to me to determine the general properties of solids bounded by plane faces, because there is no doubt that general theorems should be found for them, just as for plane rectilinear figures, whose properties are: (1): that in every plane figure the number of sides is equal to the number of angles, and (2) the sum of the angles is equal to twice as many right angles as there are sides, less four." (Biggs, Lloyd, and Wilson, Graph Theory 1736-1936, Oxford Univ Press, pp.76-77.)

A:

For a reasonably complete discussion, see Richeson's "Euler's Gem" (available on Amazon, very well written)

