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.)?
When we say "$a = b$", what we are claiming is that $a$ and $b$ represent the same object. If you apply the same operation to the same object, then you get the same result, provided that operation is well-defined.

Just to be clear, there do exist operations that are not well-defined, so sometimes you do have to prove that $a=b \implies f(a) = f(b)$. For example, take $q \in \mathbb{Q}$, represented by $q= a/b$. Then $f(a/b) := a$ is not well-defined, as $f(2a/2b) = 2a$, but $\frac{a}{b} = \frac{2a}{2b}$.
If only the operation is a single-result operation.