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?

In mathematics, you do not simply say, for example, that set $S$ exists. You would add some qualifier, e.g. there exists a set $S$ with some property $P$ common to all its elements. 
Likewise, for the non-existence of a set. You wouldn't simply say that set $S$ does not exist. You would also add a qualifier here, e.g. there does not exist a set $S$ with some property $P$ common to all its elements.
How do you establish that a set exists? It depends on your set theory. In ZFC, for example, you are given the only the empty set to start, and rules to construct other sets from it, sets that are also presumed to exist. 
In other set theories, you are not even given the empty set. Then the existence of every set is provisional on the existence of other set(s). You cannot then actually prove the existence of any set.
To prove the non-existence of a set $S$ with property $P$ common to all its elements, you would first postulate its existence, then derive a contradiction, concluding that no such set can exist. 
