You could make an analogy between the pressure distribution of a sound wave and the mass density distribution of a realistic spring undergoing vibrations, but it wouldn't give you the explanation you're looking for. As a matter of fact, that would be more like explaining a sound wave in terms of springs, rather than what you're trying to do, i.e. explaining a spring in terms of waves.
Although I'm not intimately familiar with the details, basically what goes on at the microscopic level of a spring is that, when the spring is at equilibrium, the atoms are set in some sort of rigid structure. Any given pair of atoms has a potential energy which is a function of the distance between those two atoms, so the entire spring has a potential energy determined by all the distances between every possible pair of atoms:
$$U = \sum_{i,j} U_{ij}(r_{ij})$$
In equilibrium, the spring will take a shape which minimizes this total potential energy.
If you think about it, a metal spring might typically be formed by heating some metal to make it malleable (or even melting it), and then forming it into the desired shape before it cools. The heat allows the atoms to move around relatively freely so that they can reach the equilibrium configuration that minimizes their potential energy, then once the spring cools, they are frozen in place.
Of course, the atoms are not completely frozen in place. As I see that Georg has already written in his answer, the potential energy between two atoms ($U_{ij}(r_{ij})$) has a minimum at their equilibrium distance and goes up on either side. If you add some energy into the system, say by exerting a force on it, you can get the atoms to move closer together or further apart. When you stretch or compress a string, you are really just doing this to all the (pairs of) atoms in the spring simultaneously. The atoms will, of course, "try" to return to their equilibrium position, i.e. they will "try" to minimize their potential energy, and this is what you feel as the restoring force of a spring under tension.
The word 'restoring' is synonymous with 'opposing' in that it matches the applied force, but in the opposite direction. But more so 'restoring' implies that energy is being stored - potential energy - which can subsequently be retrieved. The potential energy is the integral of force over the path of deflection:
$$E_p=(1/2)kx^2$$
The energy imparted by the pulling force is stored in the spring which is able to do work.
In it's relaxed state (position) one can arbitrarily assign 'zero' potential energy by defining 'x' as zero at that position. Any deflection relative to zero stores energy.
Another interpretation is the fact that springs tend to 'restore' position to the relaxed state once the net external forces are removed.
Best Answer
The above relation is correct. Regarding momentum, you need to consider the momentum of the entire system, including the wall, because the spring is not isolated. Just imagine the wall is very heavy but not infinitely so. And suppose both you and the wall are in a frictionless surface, otherwise there will be additional friction forces and momentum will not be conserved. In the frictionless case, when you pull the spring to the right, it will move to the right togetre with your hand, but the rest of you body will slightly move to the left to compensate (because of the reaction force), Also, the wall will move slightly to the left, minimally if it is too heavy, so the string will actually expand. If you take into account all these motions, momentum is conserved (because no external forces act on the whole system). Momentum will not be conserved if you are standing on a regular floor because friction (an external force) will act.