Why does a spring fixed to a wall have PE and not KE? As many books,including mine (Principles of Physics by Resnick and Halliday) introduce, I also will present my example like this:

Let there be a spring  of negligible mass attached on one side with a wall and free on the other side. A block of $m$ with velocity $v$ hits the free side of the spring, compressing it until its KE is 0. Then the spring will relax again to make the block move until its velocity is $v$.

Now, according to the books:

When the block was compressing, it was doing work against the spring force and its KE was converting to PE. Thus this is a case of conservative force .

Is this so? Really?
Let us think in terms of Newton's third law, and the laws of conservation of momentum and energy.
My analysis:
When the block started to compress the spring, the restoring force was opposing its movement. So, by Newton's third law, the block would impart the same force on the spring on the left. So, the block does work on the spring resulting in decreasing of its kinetic energy. So, wasn't the spring moving? Wouldn't it possess kinetic energy? If so, how should there be potential energy?
And the spring ought to have kinetic energy, otherwise the momentum of the spring-block system wouldn't be conserved. OK, if I say the wall can't be neglected, then if it had moved due to the force by the block, wouldn't the KE of the block go to the KE of the wall and of the moving spring?
So, what is going on? Why does the spring have PE? Isn't the spring moving? Can't these laws be applied here?
 A: The key is that the spring has negligible mass (that is,  zero for all practical purposes), so it kinetic energy $k=\dfrac{1}{2} mv^2$ is zero.  It is an idealization used to simplify the problem, otherwise you are correct, at least while the spring is moving, it will have both kinetic and potential energy, but calculating the last one is pretty difficult because the spring is not rigid.
A: 
When the block started to compress the spring, the restoring force was
  opposing its movement. 
  And the spring ought to have kinetic energy otherwise the momentum of the spring-block system wouldn't be conserved. Ok, if I say the wall can't be neglected, then ok, if it had moved due to the force by the block, hadn't the KE of the block gone to the KE of the wall and KE of the moving spring?

This example is similar to the one you discussed here
and the problem is not different from the general problem of elastic collisions, where a spring is inside a body (Ball): in this case the spring is not in the moving body (Block), but on the W(all). But this makes no difference, the physics doesn't change.
In an elastic collision the body is elastic: it absorbs, stores the kinetic energy as PE and then gives it back. In this example the body is rigid and the spring on the wall is compressed and absorbs the kinetic energy from B and stores in his lattices as PE.


So, by Newton's third law, the block would
  impart the same force on the spring on the left. So, the block does
  work on the spring resulting in decreasing of its kinetic energy.

As to this question, apparently there are two concepts you haven't assimilated yet, 


*

*1) the concept of negative work that I explained in your question here and now, with more details in this answer here that probably you haven't read. I will not repeat those explanations here, but,  if you carefully study those two posts, you'll just note that here the block B does positive work and the conservative spring force opposes it and does negative work on it. 


The net resulting work is the PE = 0.5 J actually stored in the spring. During this process of storage, of course, some tiny fraction of the energy would be lost and transformed into heat, I suppose we are dealing with an idealized model and example, else PE stored in the spring would be less. The spring is compressed by 25 mm, and then relaxes, returns to the position of equilibrium, pushing the block back and returning the PE stored ($\approx 0,5 J$) as KE in the opposite direction


*

*2) conservation of momentum, aka Newton's third law, (and KE) does not concern the spring as it is fixed to the wall and therefore momentum (1+1 kg*m/s) is discharged on the wall. Again, I suppose we are examining an idealized example and the wall doesn't recoil, but, if the wall were a concrete block W with, say, mass 3 tons, then, as B would bounce back with momentum -1,  W wold budge on the right (+x axis, opposite direction) with momentum +2 and v = 0.00067m/s and equal amount of KE would be subtracted to the KE of the block

A: While the spring is moving (rather it's ENDS are moving), yes they do have kinetic energy. But once the ends stop moving, that kinetic energy is lost in the form of heat (it's almost zero anyway).
The potential energy of a tense spring does not come from it's kinetic energy but the difference of force and the kinetic energy. When you apply a 5N force to pull one end of a spring, the kinetic energy of that moving end is NOT 5N. It's far less. So where does most of this applied energy (5N) go? This is stored as potential energy of the spring. When you release that pulled end, it jumps back to it's original position. It is now, when the potential energy stored in the spring is converted into kinetic energy of this angry end jumping to it's original position.
If you are a highschool student, then I'm afraid you won't understand it until you get into college :(
A: 
(a) When the (1) block started to compress the spring, the restoring [2] force was
opposing its movement. So, by Newton's third law,
(b) the block would
impart the same force on the spring on the left.


Force (1) F is lifting the box and force [2] W (weight, gravity) is opposing the motion doing negative work and subtracting energy to the box
This is from the post I deleted here. And the same principle I repeated here and it also was explained here and in the other answer: you're asking the same question over and over again.

*

*a) the forces are independent and are not [2] a reaction to (1) an action

*b) if there were a reaction, in any case it should be [2] (and not (1), the block,) the reacting force


So, the  block does
work on the spring resulting in decreasing of its kinetic energy. So,
wasn't the spring moving? Wouldn't it possess kinetic energy? If so,
how should there be potential energy?

It is the other way round: it is the spring that does negative work on the block and drains its KE and absorbs it and transforms it into heat (0.5 %) and PE (95.5%) because it is a conservative force. Even if KE were not stored, the spring can't get KE anyway,  because it is not free to move: it oscillates just a few millimeters and can never acquire velocity $v$; if the wall budges, just a tiny fracion (0.004%) of KE might be discharged on the wall.
