You seem to have the concepts a little confused. Let us sort things out:
(1) No, the words NP-hard, NP problem and NP-Complete are not synonyms of each other. NP is the class of problems that can be solved by a nondeterministic polynomial-time Turing machine.
A problem is an NP problem if it lies in NP (so, if it is solvable in poly-time by a nondeterministic Turing machine).
A problem is NP-hard if every instance of all other problems in NP can be translated to instances of it in deterministic polynomial time. Note that an NP-hard problem doesn't have to lie in NP by this definition.
A problem is NP-complete if it is both an NP problem and NP-hard.
(2) There are currently no examples found of exponential-time SAT instances, nor any proof that such examples don't exist. The whole point of the P=NP question is whether or not all problems in NP (SAT included) can be solved in deterministic polynomial time. If there were an exponential-time SAT instance, this would immediately have solved the problem in the negative. As for whether or not such a thing exists, we simply don't know as of yet.