Will Linear momentum be conserved in a non-inertial frame of reference?
In other words what is the fundamental condition for linear momentum to be conserved?
Also which is more fundamental- Newton's Third Law or Conservation of Linear Momentum? (as in Newton's third law does not hold true when we go to the quantum/atomic level or at relativistic speeds whereas linear momentum to my knowledge holds true)
Linear momentum will be conserved when the Lagrangian that describes your system is unchanged by translations in space. This is a consequence of Noether's theorem, and it's as close as you're going to get to a fundamental explanation for the conservation of momentum.
In general a Lagrangian written in the coordinate system of an accelerating observer is not going to satisfy this condition, so momentum will not be conserved.
I suspect most physicists would regard Newton's third law as derived from conservation of momentum.