Do we need a small displacement to create a oscillatory motion on the spring? Do we need a small displacement to create a oscillatory motion on the spring with a mass attached to it? Whats the limit of the displacement that we can give initally to create a oscillatory motion? Is it has to be small or it can large?
 A: The displacement sets the initial amplitude of the oscillations, which will go down from that point. So, the greater the displacement, the greater the amplitude.
The only limit here is the sterchability of the spring.
A: For an ideal spring the force as a function of position is $F=-kx$. Since this is a restoring force for all $x$, there is no limit on what initial displacement will cause oscillations (anything will do). 
Even if your spring is not ideal, the force will probably have the form of $F=-f(x)$, where $f(x)>0$ for all $x\neq0$,  so that the force is still restorative. Therefore you would still have oscillations for all displacements from equilibrium. 
Where this breaks down is in the real world your springs can't stretch forever. Even assuming perfect elasticity, your spring wouldn't be able to stretch past it's full stretched out length. For real springs, there is a point where if you stretch the spring past it, the spring will not restore to its original form. Unfortunately, this depends on the physical spring itself, so at this point I can't keep talking in general.
