Some questions on stereochemistry
1) Why does a molecule become achiral if it has a plane of symmetry
  even though it has stereocenters?


If a plane of symmetry bisects a molecule such that one half of the bisected molecule is the mirror image of the other half of the bisected molecule, then the entire molecule must be superimposable on its mirror image, it must be achiral.  Said differently, if mirror symmetry exists within a molecule, then the mirror image of the entire molecule will be superimposable upon the original drawing of the entire molecule.

Here's an example of a molecule ($\ce{A}$) with 2 chiral groups ($\ce{G_{R}}$ and ($\ce{G_{S}}$).  There is a plane of symmetry that bisects $\ce{A}$, it is perpendicular to the plane of the screen and contains the central carbon atom and the 2 methyl groups.  Now, when we draw the mirror image of $\ce{A}$ (let's call it ($\ce{\bar{A}}$), we find that it is superimposable upon the original drawing of $\ce{A}$ by a simple 180-degree rotation.
Even though $\ce{A}$ contains 2 chiral substituents, the molecule as a whole is achiral.