A key concept for understanding fluid mechanics is ergodicity. Informally, when you assume ergodicity, you assume that all of the little transient details about the system wont affect the final outcome, because the statistical behaviors are all that will matter (the actual definition is more nuanced than that, but that informal definition should be enough for answering this particular question). In the case of your static fluid, you are correct that the particles in the fluid are not at rest. They're bouncing around with thermal energy, sometimes at quite high velocities. However, in a static fluid, if you look at average motion, its 0. For every particle that's traveling to the left, it's statistically likely that there's a nearby particle traveling to the right.
As you noticed, this is the cause of pressure on objects. IF you look at any boundary on a fluid, you'll have fluid particles bombarding it from one side (with some statistically expected velocities), and no fluid particles on the other side, so you can easily see how this can impart impulse, which can be related to pressure.
In your equilibrium example, Wikipedia is not perfectly correct in their statement, but they're very close. In many fluid situations, you have the force of gravity pulling down on all fluid particles. To be in equilibrium, the pressures on the bottom side of your cube actually have to be slightly higher than the pressures on the top. The difference is exactly enough to counteract the extra downward force that the cube of particles applies due to the force of gravity on its mass. Now if the cube is really small, the difference here becomes vanishingly small. However, it's nonzero, and if you are integrating pressures, those little tiny differences can add up. In fact, when you look at the concept of buoyancy, you'll find those tiny differences add up to exactly the buoyancy force.
This minor detail aside, the pressures in all directions on that cube must be nearly equal. The reason for this is that the system will equalize itself if they are not equal. Consider a case where the left and right forces are substantially lower than those on the top and bottom (and assume constant temperature to make the description easier). Nothing stops the particles from flowing into and out of the cube, and the rate at which they flow is proportional to the pressure in that little region of fluid. In this example, lots of particles will flow into the cube from the top and bottom, very few will flow in from the left and right. Meanwhile, the same number of particles will flow out of the cube in all direction, based on the pressure in the cube. This will raise the pressure in the cube until the net movement of particles coming in from the top and bottom is equal to the net movement of particles going out through the left and right.
This will generate a small flow, redistributing particles from the top and bottom through the cube into the left and right regions. This process continues until the pressures are all brought into statistical equilibrium. At this point, there is no statistically significant movement of particles (they're all moving around equally in all directions), so the fluid is now "static." This is why the pressures in all directions are nearly equal in a static fluid: because the only state where the net fluid flow is 0 is when these pressures are all nearly equal!