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It's not very useful to think of the frequency as negative. The wave vector $\vec{k}$ should be interpreted as the indicating the direction of the wave, so if the wave is going in the opposite direction, then $\vec{k}$ should be made negative, not $\omega$.

Here is an animation that may help you. It shows the two propagating waves, and the resulting standing wave. It's called a "standing" wave in the sense that the nodes (places where the wave function has a value of zero) of the wave do not move.

It's a little odd to say you have standing waves that are propagating, but yes, you will have a two dimensional standing wave, with a different wavevector in each dimension. Here is a visualization of it frozen in time (I used $|\vec{p}| = |\vec{q}| = 1$ for simplicity).