if (dis)proving a conjecture on graph theory can be done just by a counter example then can every (dis)proof be mapped actually to a counter-example? Suppose we have a conjecture on graph theory that can be (dis)proved by means of a counter example, then, is it true that every alternative (dis)proof of the conjecture can be mapped to a counter example? 

This is in the general case, but for instance, can any proof that the hadwiger's conjecture is false be mapped to a counter example, i.e., a particular graph? 

Or, 

Can any proof about a purported property $P(L)$ of a language $L$, susceptible of being (dis)proved by a counter example, be mapped to a particular word, say $w$ meaning $w$ a counter example, i.e., $P(L):=FALSE\;$  since $w$ exists?
Your question isn't entirely clear, but it seems you're asking if every false graph property has a counterexample. Obviously this is true: by definition there must be some graph for which the property doesn't hold. On the other hand, a proof that the property doesn't hold for all graphs may not necessarily give an example of such a graph; this is called a non-constructive proof.