Is it right to say this regarding Newton’s third law of motion?

I have named the forces 1 and 2 now suppose you are applying force 2 on the wall and wall 1 is fixed.

First of all Spring force would not act until there is a extension in spring. So If you pull a wall attached to spring then the spring force would increase according to th law $$F=kDelta x$$ where $k$ is spring constant and $Delta x$ is extension in spring

So it will move until the spring gets enough elongated to have ability to cancel the effect of F1 you are apllying.

And for the case of wall 1(fixed) force will also increase on it but the friction or (some other mechanism) would increase the force and remain equal to the spring force everytime So the equation you are applying is at the moment of maximum elongation.

So at that moment the wall will not move but physically you are feelingthe situation at the moment when the spring is relaxed.

Supposing that something is not moving, then it won't move unless an unbalanced force is acting on it. Generally for a wall you can pull on it and there will be other forces (from the ground, the rest of a building, whatever) which will balance your pull, and it won't move. You could have a movable wall where you can pull it harder than the other forces - in this case the wall will move.

Suppose you have a spring trying to pull a movable wall towards a fixed wall. If the force in the spring is strong enough it will be stronger than the other forces on the movable wall so that wall will move, while the other forces on the fixed wall will exactly balance the spring force on that wall and the fixed wall will not move.

There are a number of action-reaction pairs (according to Newton's 3rd Law).

1. Spring on movable wall and movable wall on spring. ($F_{ext}$)
2. Spring on fixed wall and fixed wall on spring. ($F_{ext}$)
3. Ground on movable wall and movable wall on ground. ($F_G$)
4. Ground on fixed wall and fixed wall on ground. ($F_{ext}$)

We also need to consider the pairs of forces acting on each object.

A. Movable wall. A larger force ($F_{ext}$) from the spring inwards and a smaller force ($F_G$) from the ground outwards. Because there is an unbalanced force this wall begins to move.

B. Fixed wall. An inward force ($F_{ext}$) from the spring, and an equal outwards force ($F_{ext}$) from the ground. Because the forces balance this wall does not begin to move.

C. Spring. The tension ($F_{ext}$) in the spring pulls roughly equally on both ends. The forces at the ends are not exactly equal so the spring begins to move, but the difference is too small to measure easily.

D. Ground. There is an inward force ($F_{ext}$) at the fixed end and a smaller inward force ($F_G$) at the movable end. The ground will begin to move but because the earth is so heavy nobody will notice the movement.

The result of all this is that the movable wall (and the spring) move towards the fixed wall, while the fixed wall and the Earth do not appear to move at all.

Many people get confused by Newton's action/reaction forces. The situation is not helped by the expression "reaction force" having another meaning as well.