Q1. So what is the "simplest example" of a compact complex manifold of dimenension $2$, say $X$, for which $H_B^2(X,\mathbb{Z})$ has nontrivial torsion.
Q2. How do we think about these torsion elements? What is the geometrical content behind it?
Q1. So what is the "simplest example" of a compact complex manifold of dimenension $2$, say $X$, for which $H_B^2(X,\mathbb{Z})$ has nontrivial torsion. Q2. How do we think about these torsion elements? What is the geometrical content behind it? 


The torsion of $H_B^2(X,\mathbb{Z})$ is that of $H_1(X,\mathbb{Z})=\pi_1(X)^{ab}$ (universal coefficient theorem for cohomology), so the simplest case should be a simply connected complex surface quotiented by a fixed point free holomorphic involution (or a prime order automorphism). I would propose an Enriques surface, but I'm not at all convinced it is "simplest". 


Are you interested in a complex manifold which is not Kaehler? $\mathbb RP^3\times S^1$ admits a complex structure. Remove the origin from $\mathbb C^2$, and then divide by the free action of $\mathbb Z\times \mathbb Z/2$ generated by $(x,y)\mapsto (2x,2y)$ and $(x,y)\mapsto (x,y)$. 

