Most non-physicists have a sufficiently murky definition of the concept of “mass” that it’s impossible to say if the things conceived of are conserved. And I suspect that of the few non-murky cases, the definition will turn out to be equivalent to the one that physicists use (though the layman might not realize that, nor its implications).
And yes, conservation of mass can be derived from the combination of conservations of energy and of momentum, since together those comprise the four-momentum, and mass is just the norm of four-momentum.
Conservation laws are related to symmetries by Noether’s theorem, so for each symmetry there is a conservation law. Conservation of momentum is related to space-translational invariance and energy to time-translational invariance and are true both in Newtonian physics and special relativity, the difference being that in special relativity they can be both seen as part of spacetime-translational invariance (leading to conservation of 4-momentum). Mass conservation in Newtonian physics is trivial and exists by definition, in special relativity the sum of masses are not conserved and the conservation of total rest mass is just a result of spacetime-translational invariance.
Rotational-invariance in space leads to the conservation of angular momentum, however there is no conservation of rotational energy and boost-invariance leads to a completely useless conservation law.