When is conditional Kolmogorov complexity zero?
The answer depends on whether your encoding is prefix-free or not, and in the latter case, on your universal Turing machine.

There are two variants of Kolmogorov complexity: one in which the set of programs form a prefix-free code, and one in which there is no such requirement. For the first variant, the Kolmogorov complexity is always positive. For the second variant, the universal Turing machine can do whatever it wants given an empty program (or indeed, any fixed program). You can arrange that $K(x_1|x_2) = 0$ for your choice of $x_1,x_2$, for example.