I know this question may seem a bit laughable, but the way the equations for general relativity are formed is through Poisons equation: $$\nabla^2\phi=4 \pi G \rho$$ Which are formed using Newton's law of gravity? You may say 'General relativity describes curved space-time through the metric and Newtonian gravity describes a vector field". However, wouldn't that be equivalent? If we draw an axis of space and time and draw an accelerated world-line and made it straight, it would look like space-time is curved. So why is Newton's law different to Einstein's laws, if we can describe the phenomena through 2 different ways? Is it the energy-momentum tensor? When we use the geodesic equation I interpret that as changing curved space-time into a vector field to describe the motion relative to flat space-time, why wouldn't that vector field be equivalent to newtons laws? Sorry if these question seems trivial, I'm just confused that's all.
Why isn't general relativity equivalent to Newtonian gravity?
Poisson equation alone does not allow for relativistic causality.
An important ingredient of relativistic theory is the existence of finite speed of signal propagation. Poisson equation contains only spatial derivatives, so any changes in the sources are instantaneously reflected in the potential, and if the potential alone determines some observables of the theory we would have instantaneous propagation.
Note, that it is still possible to have Poisson equation and relativistic causality if the theory also has additional degrees of freedom that propagate relativistically and that observables of the theory are gauge invariant functions of the potentials. This is the case with electrodynamics in Coulomb gauge. The equation for scalar electrostatic potential is precisely the Poisson equation but there is no causality violation, since contributions from the vector potential would compensate for the effects of instantaneous propagation of electrostatic potential.
Similar situation could occur in (linearized) theory of gravitational field: by imposing appropriate gauge condition one could have Poisson equation for a specific component of the gravitational field (which we could identify with Newtonian gravitational potential), but the theory must also have additional degrees of freedom independent on this one potential to retain relativistic causality.
So just knowing from astronomical observation that Newtonian gravitational theory describes the Solar system really well and postulating principles of relativity we would inevitably come to the conclusion that the theory must be relativistic theory of spin–2 field (spin–0 is eliminated by absence of gravitational aberration established by Laplace with high degree of precision, while for spin–1 field like charges would experience repulsion rather than attraction), i.e. general relativity.
I’ll answer your first question. You are supposed to ask one, not six.
Why isn’t General Relativity equivalent to Newtonian gravity?
Poisson’s equation is linear in the potential. Einstein’s equations are nonlinear in the metric. There is no mapping that could make them equivalent, because they don’t even have the same number of degrees of freedom. (The potential is one number at each point; the metric is ten numbers at each point.) However, Einstein’s equations do reduce to the Poisson equation in the weak-gravity limit.