How exactly do we superimpose molecules while checking optical activity? I am struggling with these type of questions where in some cases we rotate a molecule's mirror image by 180 degrees to check superimposability.  

Like when checking any tetrahedral chiral compound, we draw its mirror image and rotate it by 180 degrees and say that it's not superimposable. 
Suppose the mirror image of the left-handed version is the right-handed version, hence both are not superimposable. If we rotate the right-handed version by 180 degrees and then try to check, will it be superimposable now? Similarly, trans 1,2 dimethyl cyclopentane is chiral but if we rotate its mirror image by 180, is it superimposable?
You're being confused by the rotation. You take the mirror image. Any rotation or sequence of rotations that superimpose the mirror image over the original means that the molecule is achiral. It's not just a 180-degree rotation.