Momentum is conserved in every collision; kinetic energy is conserved only in elastic ones. Why does one survive an inelastic collision and the other not?

B. Because internal forces cancel, while kinetic energy can become heat.

Momentum conservation follows directly from the third law — the internal forces are equal and opposite, so the two momentum changes cancel — and nothing about the collision's messiness touches that. Energy is still conserved in total; it simply moves into heat, sound and permanent deformation. Which is why an elastic collision gives a second equation and can be solved completely.