Pairing - Is it possible to map two $r$-torsion points to a $r^2$-torsion point?
No, it is not possible.

By the definition of bilinearity, we have $e( kG, H ) = k \cdot e( G, H )$.  If the order of $G$ is $r$ (that is, $rG = 0$, we have $e( rG, H ) = e( 0, H ) = r \cdot e(G, H)$.  We know $e(0, H) = 0$ (as bilinear functions maps the identity to the identity), and hence we have $r \cdot e(G, H) = 0$; that is, the order of $e(G, H)$ must be either $r$, or some divisor of $r$.