The original poster believes to having found contradictions in Newton's 3 axioms. IMHO, the contradictions stem from his personal understanding aka interpretation of Newton's text. Let me present an understanding being free of those contradictions.
The first axiom represents Galilean inertia and...
As a schoolboy, I was highly excited about the fact, that so many physical content can be formulated in the most simple form \delta S = 0. Much later I learned, that Lagrange's and Hamilton's representations of classical mechanics treat position and velocity/momentum variables on equal footing...
The PLA is a 'principle', not a 'fundamental law' or 'axiom' like Newton's axioms. It is usually equivalent to the equation(s) of motion. Now, in many cases, the equation of motion is not fundamental. Example 1: Newton's equation of motion, force = mass x acceleration. It does not enter Newton's...