Depth adjustment definitely affects the opening as deeper means the blade has moved forward on the geometric plane defined by the angle of the frog relative to the sole of the plane. This shortens the distance between the cutting edge and the front of the mouth opening, that is, the mouth has become narrower. Lateral adjustment does not move the cutting edge forward; it simply rotates the blade on that single geometric plane so that the cutting edge is evenly projecting from the mouth at the same depth. The OP said using the lateral adjustment lever made the mouth opening "off" or uneven. I suppose if you were using your lateral adjustment lever to set up the plane to have one side take a significantly heavier shaving than the other (something I do occasionally, but mostly with spoke shaves, not planes), you are effectively making one side deeper than the other and have a slightly different mouth opening (narrower on the deeper side).
~ Do not seek to follow in the footsteps of the men of old; seek what they sought.