Applying force in zero seconds? How is this possible?

Beware of blindly applying equations.

In thinking about the velocity of the wall, then for $F\Delta t=m\Delta v$, $F$ is the net force acting on the wall. If $\Delta v=0$, then we know that $F=0$ (and the same is true vice versa).

Note that this is just Newton's second law: $F=m\frac{\Delta v}{\Delta t}=ma$. If $a=0$ then we can conclude that the net force $F$ must be $0$.

Also, if we really did have $\Delta t=0$, then of course $\Delta v=0$ since nothing can change over no time.

the only way the equation equals zero is that Δ equals zero, applying force in zero seconds?

No. The force $F$ is the net force acting on the wall, not the force you apply to the wall. So the net force can be zero.

You apply a force $F$ to the wall. The floor that holds the wall in place applies an equal and opposite force of $F$ on the wall, for a net force on the wall of zero and no acceleration of the wall per Newton's second law $F_\text{net}=ma$.

At the same time you apply a force $F$ to the wall it applies an equal and opposite force $F$ on you per Newton's third law. But you do not accelerate because the static friction force between your feet and the floor is equal and opposite to the force the wall exerts on you, for a net force on you of zero.

Hope this helps.