Is there a proof for the fact that, if you perform the same operation on both sides of an equality, then the equality holds? Is there a proof for this, or is it just taken for granted? 
Does one need to prove it for every separate case (multiplication, addition, etc.), or only when you are operating with different elements (numbers, matrixes, etc.)?  

No, it needs no proof. In general one has that if $x=y$ and $f$ is a function that has $x$ and $y$ in its domain then $f(x)=f(y)$. Since $a+b$ is just another way of writing $+(a,b)$, one has that if $a=c$ and $b=d$ then $a+b=c+d$ (same for multiplication).
