>Under normal, everyday circumstances, Newton's laws still maintain their validity, but under certain extreme conditions, they have to be replaced by more accurate formulae, from the Theory of Relativity, or from Quantum Physics.
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>This brings up the question, what parts of Newton's theories are still considered valid, and what parts not.

Newton's mechanics is always invalid, but...

Try to calculate the same thing under Newton's formulae, and under relativistic formulae. The difference will appear somewhere on the 6th significant digit, or even lower than that - depending on the speed. So it's wrong, but the error is below our radar, for most of practical uses.