Lets start from the beginning: Physics theories are mathematical models that fit existing measurements and observation and, very important predict correctly new measurements and observations.
In order to pick from the plethora of mathematical solutions those that can be used to consistently fit the data extra axiomatic statements are used, as strong as the mathematical axioms. These are called laws, postulates,principles and are always true.
The law of conservation of energy is one of these , and together with the conservation of momentum and angular momentum are mainstays of all branches of physics.
The progress in experimental physics and the need to keep energy conservation as a law has led to special relativity and the algebra of the four vectors, necessary to understand all atomic, nuclear and elementary particle phenomena. Our present day cosmology has defined dark energy and dark matter in order to keep the conservation of energy law true, which means that maybe for general relativity , large masses and energy a new look might be needed on the energy conservation law, but your question is not at this level.
For classical kinematics, conservation of energy has to take into account all forms of energy in the summation,in the system under observation. There is no separate conservation of kinetic energy, potential energy,heat energy...
If talking of matter, thermodynamic forms of energy have to be considered in the summation. If the scattering is inelastic, part of the kinetic energy becomes potential energy (sping type deformations), or even heat (permanent deformation) and radiation energy, in the case of extended objects. All these depend on the constants of the material , see for specific heat for example.
Edit after your edit.
Your first equation does not follow the momentum conservation law, because you divide by the input mass. The units are also wrong, left and right side. Conservation of momentum between two interacting masses is simply $m_a*v_a=m_b*v_b$ $v$s vectors, $p_a$ before the interaction $p_b$ after. The $p$ vectors.