Does everything in this world vibrate? Why? I was introduced with something called natural frequency of a body. It was taught us in the chapter Simple Harmonic Motion. Our teacher said that everything in this world has its natural frequency of vibration. And if we attach it to a oscillator with same frequency then due to resonance the amplitude will go on increasing and at some time the amplitude will increase to such a extent that the system will collapse. 
So Why do everything vibrate? We do not notice it through our naked eyes.. Does it really even vibrate. Even if it is at microscopic level.
This means we do not need any wrecking balls for demolishing the walls. Just get a oscillator .. Fit it to both sides of the wall and Boom! Wall's demolished. 
P.S.: I haven't seen an oscillator, nor have I seen a picture. I just picture it as a periodic force generating machine. 
If everybody vibrates .. then why don't we destroy things this way?
 A: You are talking about simple harmonic motion, so my answer will focus on that and ignore molecular vibrations (that doesn't seem to be what you're talking about).
For starters, I think it's good to consider amplitudes.  Although everything has a natural frequency, the amplitudes of vibration are usually very low, so much so that observing them is difficult.  This occurs because in reality there is a dampening which decreases the amplitude of oscillation with time (due to air resistance and other dissipating effects).  This means that if a damped oscillator is left on it's own it will have an amplitude close to 0, so we don't notice it. An ideal simple harmonic oscillator (no damping) would maintain it's amplitude forever if there are no external forces.
Regardless of if the amplitude decays, there is still a frequency of the oscillations.  When left to oscillate on it's own (no external applied force), the frequency will be the "natural frequency".  If you were to apply a periodic force with the same frequency, you will create "resonance".  This can be explained through superposition.  In the case of resonance you constantly have a steady amplitude that matches the systems amplitude, so the driving force keeps adding energy to the system, and the interference is always constructive.
On paper, without damping, the energy would increase forever.  In practice this energy will eventually make something break, or just dissipate itself.
As far as your wrecking ball thing goes, you could do that.  One problem will be controlling the vibration so it only effects your building, not other buildings in the area.  Another problem would be controlling the collapse, it may be too unstable when resonating.  A wrecking ball controls where you want to break next, doing so with resonance would require a lot of planning and may not be possible depending on how the system responds.
A third (and probably prohibitive economically) problem is the required force you need to oscillate the structure with to overcome its self-damping.  A very small force would be easily damped out in the regular movement of the system.  Maintaining a high-energy oscillation is quite unsafe; might as well use a wrecking ball or explosives. 
A: Unless you're at a temperature of 0 Kelvin (~ -273 degrees, "Absolute zero"), things with mass will have kinetic (movement or vibrational) energy - note that at 0K they will still possess internal energy, such as binding energy.
The vibration will be negligible to the naked-eye.
Natural frequency in your case will be to do with larger systems (not microscopic), such as a car driving over equally spaced humps and jumping higher each time, or a bridge being pushed by the wind.
For example, your can break a wine glass using a specific (the natural) frequency of sound, that will cause the structure to resonate.
You could probably similarly break down a wall using this technique, however, it may simply be cost/time effective to use something like a wrecking-ball.
Another is this bridge that collapsed due to it being forced by the wind to resonate, until it could no longer stand the stress. https://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_(1940)
Today, systems will incorporate forms of dampening, to limit or stop resonance.
A: Nikola Tesla was working on some ideas like this.  
Tesla's oscillator - Wikipedia
The Mythbusters tried out this idea on a bridge, and although the results were a bit impressive, they weren't close to the dramatic result that Tesla had proposed.
http://www.discovery.com/tv-shows/mythbusters/about-this-show/earthquake-machine/
Tesla also had the idea that he could induce earthquakes with multiple carefully timed explosions.  Aside from an episode of Wonder Woman (The Pluto File), I haven't seen too much discussion about this idea, aside from conspiracy theory websites.
https://en.wikipedia.org/wiki/Tectonic_weapon 
But realistically, the damping forces of most objects are so strong that we are typically in the "overdamped" situation, where things don't really oscillate at all.
https://en.wikipedia.org/wiki/Harmonic_oscillator#Damped_harmonic_oscillator
