1) Diffraction will be technically unaffected - the effective aperture is reduced because light from the central portion of the lens' image circle is spread over a larger area. But, the physical aperture (iris disphragm diameter) of the lens is unchanged, and that's what determines the amount of diffraction. But...in practice, the effect of diffraction will be more evident for the same reason - the light is spread over a larger area. So, in practice, you can think of the TC as affecting diffraction - in other words, if adding a 2x TC to an f/5.6 lens on a 7D, the resulting image will be softer due to the effects of diffraction since f/11 is well over the DLA for an 18 MP APS-C sensor.
2) DoF will deeper with the TC (for the same framing), because you're comparing f/5.6 vs. f/11 at the same framing. If you didn't move, you'd have a shallower DoF with the TC due to the longer focal length (and a smaller FoV, obviously).