> why massless?Meaning, why is the spin-2 field massless? The easiest short answer is, because gravity is a long range interaction, and only massless gauge bosons can give you a long range interaction. That's why electromagnetism is also long range (the photon is massless) but the weak interaction is not (W+, W-, and Z are massive).
The strong interaction, unfortunately, breaks this simple heuristic (the gluon is massless but the strong interaction is short range), because it has other factors involved which aren't as easy to explain. But we know that those other factors don't come into play with the spin-2 field model of gravity, so we can still safely say that the graviton should be massless.
> If I understand GR correctly, the gravitational field itself (especially in the form of a gravitational wave) has energy
Only in a certain sense, which is...
> and therefore contributes to the mass-energy tensor, further bending spacetime.
...not this sense. The stress-energy tensor does not contain any "gravity" contribution. In the Einstein Field Equation, "gravity" is on the LHS, represented by the Einstein tensor, and "mass-energy" is on the RHS, represented by the stress-energy tensor. That has to be the case in order for local conservation laws to hold.
What all that means is that there is no valid local concept of "energy stored in the gravitational field" (because there is no tensor quantity that corresponds to this). But globally, you can still look at a system that emits gravitational waves and come up with a meaningful concept of "total energy" that decreases with time as gravitational waves are emitted. (This concept is called the "Bondi energy".) So we can say that "gravitational waves carry energy" in this sense. We just can't localize where that energy is.