To answer your second question, the undisturbed fluid, liquid or gas, is under a constant pressure in equilibrium, say P0. The acoustic wave is a change in this background value, say delta(P) not a differential but a small perturbation. The wave equation can be derived from first principles using the equations of fluid mechanics by assuming small perturbations of all macro variables about their equilibrium position.
To elaborate on your first question, answered by another post, the pressure wave is the solution of the wave equation that describes the behavior of the small perturbation in pressure, delta(P). The "wave" will travel through the medium despite the elements of the medium staying in their local neighborhood. The displacement of particles from their equilibrium position causes a momentary increase in density in the direction of motion and a decrease of density in the opposite direction. This in turn generates the pressure difference. Collisions with more particles will cause the initially displaced particles to move back to their equilibrium position and a new pocket of fluid particles will travel in the direction of the displacement. The initially displaced particles are now in equilibrium (more or less) and the disturbance moves in the direction of the initial displacement. If there is a vibrating source then the process continues at the frequency of the vibrating source.
Some descriptions of acoustics do not directly solve for pressure waves, but do solve for the displacement potential, or velocity potential. The equations of state relate pressure, density, and this potential function so once you have one you have them all.