No, inertia is motion and motion is energy and energy is mass.

To get energy from a motionless object you have to extract the energy from it's mass, thus converting mass into energy. The matter is not annihilated, merely transformed.

It's the same principle as with energy and motors. The mechanical energy of a turbine is turned into electrical energy and the electrical energy is turned back into mechanical energy.

The same applies to energy (E), mass (m) and motion (c^2). Matter has mass except for a few (photons, gluons and gauge bosons) which in turn have to always move at the speed of light to make up for the lack of mass and thereby upholding the equation.

E=mc^2 accounts even for these exceptions (which Einstein didn't even know about at the time), which is why it's so important.

To answer your questions, yes mass is a property of matter. Matter has mass and volume and therefore is defined as matter. It's like asking of "trueness" is a property of a boolean: Yes it is, since a boolean is defined by having trueness and falseness. If it didn't have trueness (thus only having falseness) it would not be defined as a boolean anymore.

It's mass is a property of matter by definition, because if a "something" didn't have mass you couldn't define it as matter anymore. Same thing would work in reverse, if something had matter but no volume it wouldn't be matter either.

And as to expand on "The matter is not annihilated, merely transformed.": You have matter(mass + volume) of which you transform the mass into energy, and you are left with volume but no mass and since a "something" without both mass and volume is not matter anymore, the matter "ceases to exist" which can be misunderstood, but merely means that you have taken the mass away and what is left cannot be defined as matter anymore. It is not, in fact, annihilated.