What does it mean for a set to exist? Is there a precise meaning of the word 'exist', what does it mean for a set to exist?
And what does it mean for a set to 'not exist' ?
And what is a set, what is the precise definition of a set?

There is no definition of what a set is. The reason is simple: it is hard to imagine anything more fundamental than a set that you could use in order to say what a set is. And if there were such a thing, let's call it a tet, then one would ask "but what exactly is a tet? can we define it in terms of something more fundamental, perhaps in terms of a uet or a vet or a wet or xet or a zet?". This never ends. 
So, what we do is we take one of two approaches. The naive approach resorts to trying to brush the vagueness away by saying things like "a set is a collection of things where order is not important, and repetition is not possible". While not incorrect, what the hell is a collection then? Well, in the naive approach we don't care and just assume we all share the same intuition as to what sets are. This works up to a certain degree. 
The other approach is the axiomatic one. Instead of saying what sets are, we say what we can do with them and which laws govern them. There are many different axiomatizations of set theory, none of which is particularly simple (e.g., requires a bit of effort to understand). Fundamental results in axiomatic set theory include such results as "it is impossible to prove the consistency of set theory", essentially saying you can't be sure that a universe of sets as we pretend to exist really exists. Other results show that certain fundamental axioms are independent of others, showing that what one may consider 'obvious' about her idea of sets need not be so obvious for others. 
