This is not a complete answer to your question, but it may give you some insight… Consider a head-on collision, observed in the laboratory frame of reference, between two spheres of equal mass. There will be another frame, known as the centre of mass (CM) frame, in which the same collision is seen as the spheres approaching each other with equal and opposite velocities.
For example suppose the spheres have lab-frame initial velocities in the x-direction of $u_1$ and $u_2$. The CM frame is the frame moving in the x-direction with velocity $\frac{u_{1}+u_{2}}{2}$. In this frame the spheres will have initial velocities $\frac{(u_{1}-u_{2})}{2}$ and $\frac{(u_{2}-u_{1})}{2}$ [Galilean transform].
Now, whether the collision is elastic or inelastic, it is a matter of symmetry that in the CM frame the spheres' velocities after the collision will be equal and opposite. Call them $v_{CM}$ and $-v_{CM}$. If the collision is inelastic these velocities will be of smaller magnitude than the initial velocities in the CM frame. In the laboratory frame these velocities will be $v_{CM} +\frac{u_{1}+u_{2}}{2}$ and $-v_{CM} +\frac{u_{1}+u_{2}}{2}$. So the vector sum of 'final' velocities in the lab frame is $u_{1}+u_{2}$, that is momentum is conserved!
Note that we've achieved this result by using little more than symmetry and the Galilean velocity transform between reference frames. Clearly things aren't quite as simple for bodies of different mass, but the message is the same: conservation of momentum can be regarded as a consequence of a spatial symmetry.
We note that the argument depended on the vector nature of momentum (or of velocity). The same argument could not be used for the scalar, kinetic energy.
