Are there any natural $\Pi_2^P$-complete problems?
There are very many natural complete problems for $\Pi_2^p$, and there is a survey [1] on completeness for levels of the polynomial hierarchy, containing many such problems.  The paper On the complexity of min-max optimization problems and their approximation [2] contains a nice overview of "min-max problems" with several proofs of completeness.  The latter paper is opened   with the following sentence:


  The computational complexity of optimization problems of the min-max form
  is naturally characterized by $\Pi_2^p$, the second level of the
  polynomial-time hierarchy.


Some problems:
Here are some examples, all of which are $\Pi_2^p$-complete, listed in the aforementioned survey [1].


$\forall \exists \text{3SAT}$: Given 3-SAT formula $\phi(x,y)$, is it true that for all $x$ there exists a $y$ such that $\phi(x,y)$ is satisfiable? 
NOT-ALL-EQUAL-$\forall\exists\text{3SAT}$
MINMAX SAT, MINMAX CIRCUIT, MINMAX CLIQUE
LIST CHROMATIC NUMBER
GRAPH SATISFIABILITY
DYNAMIC HAMILTONIAN CIRCUIT, LONGEST DIRECTED CIRCUIT
SUCCINCT TOURNAMENT REACHABILITY
CONSTRAINTS OVER PARTIALLY SPECIFIED FUNCTIONS
ARGUMENT COHERENCE
3-COLORING EXTENSION, 2-COLORING EXTENSION
(STRONG) ARROWING, GENERALIZED RAMSEY NUMBER
etc., etc.


References:

[1] Schaefer, Marcus, and Christopher Umans. "Completeness in the polynomial-time hierarchy: A compendium." SIGACT news 33.3 (2002): 32-49. (PDF)

[2] Ko, Ker-I., and Chih-Long Lin. "On the complexity of min-max optimization problems and their approximation." Minimax and Applications. Springer US, 1995. 219-239. (PDF)