Mass has the same value in all inertial reference frames? Is  mass the same in all inertial frames? If it is, why is that? If not, can you also explain?
 A: Of course it is, that's why it's called "rest mass". Every body is at rest in his own inertial frame. In other inertial frames relative to which your mass is moving it also has kinetic energy, and since energy and mass are equivalent there is also the so called "relativistic mass", but when you only say "mass" you usually mean "rest mass" and that is invariant by definition.
A: Mass is 100% about the balance between energy and momentum: $$mc^2=\sqrt{E^2-(\vec pc)^2}.$$
For some systems, the mass is zero. For others it is nonzero. When it is nonzero, the the energy momentum vector points in the same direction as the tangent to the worldline of the particle. And further the energy momentum vector is nothing other than $mc^2$ times the unit tangent to the worldline.
Everyone agrees on the worldline (it's the set of events where the particle is) and everyone agrees on the  tangents to the worldline. And everyone agrees if you have a non null vector you can make a unit vector out of it. And everyone agrees that unit vector and the energy-momentum vector point in the same direction and since $E^2-(\vec pc)^2$ is the squared length of the energy-momentum vector, its length is just $mc^2$.
So we know what mass is, it's the (possibly null) length of the energy-momentum vector, an actual vector in Minkowski spacetime. With an actual length.
So there simply isn't anything to disagree about. The length of a tangent vector doesn't depend on frame, and neither does the length of the energy-momentum vector.
A: "Why" is a often tough question to answer satisfactorily  in theoretical physics. In this case, mass is constant in inertial reference frames because that's part of how they are defined by special relativity:

Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.

— Albert Einstein: The foundation of the general theory of relativity, Section A
In special relativity, "mass" generally refers to the rest mass (or invariant mass) of an object, which is the Newtonian mass as measured by an observer moving along with the object.
The rest mass of a particle in inertial frame A will hold in in all inertial frames, as that's the entire point of inertial frames - they do away with the relativistic effects of acceleration.
