Is this result wrong because of calculation time? (and more questions about Assumptions)

Question

I am confused with the Integrate Given by Mathematica. First let's see a one-dimensional case:

    Integrate[UnitStep[k - Sqrt[kx^2]], {kx, -Infinity, Infinity}, Assumptions -> k > 0]

The result is 2k. Right answer! You can change UnitStep to HeavisideTheta, it still gives the same correct answer.

However, when changing to three dimension case,HeavisideTheta gives the wrong answer. See the following:

Integrate[ HeavisideTheta[k - Sqrt[kx^2 + ky^2 + kz^2]],
  {kx,-Infinity,  Infinity}, {ky, -Infinity, Infinity}, {kz, -Infinity, Infinity},
  Assumptions -> k > 0]

Mathematica gives the wrong answer 0 and with an error message I don't understand.

Refine::fas: Warning: One or more assumptions evaluated to False.

If Assumptions is not added, it directly gives 0 without any WARNING! As if it calculated it right! How dangerous if I previously didn't know the answer!

@Jens suggested using UnitStep instead of HeavisideStep. I tried: in Mathematica 7 it fails (giving the wrong answer 0) but in version 8 it gives the right result:

    (4 k^3 \[Pi])/3

Still with the same message:

Refine::fas: Warning: One or more assumptions evaluated to False.

So what does this message mean? Why do different Mathematica versions give different answers? Is this because of the computing time involved?

---

Above is the main question, here is some subtle question I would like to ask; see this integral:

Assuming[k > 0, Integrate[ UnitStep[k - Sqrt[kx^2 + ky^2 + kz^2]],
  {kx, -Infinity, Infinity}, {ky, -Infinity, Infinity}, {kz, -Infinity, Infinity}]]

the results:

    (4 k^3 \[Pi])/3 k>0
     0  True

It seems Mathematica 'forgets' that I have assumed that k>0. Why is that? What does the second line of the result mean?