Do we know if there exist true mathematical statements that can not be proven? Given the set of standard axioms (I'm not asking for proof of those), do we know for sure that a proof exists for all unproven theorems? For example, I believe the Goldbach Conjecture is not proven even though we "consider" it true.
Phrased another way, have we proven that if a mathematical statement is true, a proof of it exists? That, therefore, anything that is true can be proven, and anything that cannot be proven is not true? Or, is there a counterexample to that statement?
If it hasn't been proven either way, do we have a strong idea one way or the other? Is it generally thought that some theorems can have no proof, or not?


do we know for sure that a proof exists for all unproven
  theorems?...Phrased another way, have we proven that if a mathematical
  statement is true, a proof of it exists?

Perhaps this is a semantic issue, but I don't think these two questions are identical, even though you may have intended them to be so.
If a mathematical statement has been proven as true, and the proof is correct, then yes we have proven that a proof exists. It doesn't mean that it's the only proof possible. In the book "Proofs Without Words" there's around 13(?) proofs of the Pythagorean Theorem with no words at all, just diagrams.
For statements that haven't been proven yet, there are two possibilities. Somebody can prove that the statement can't be proved by pointing out a flaw in the reasoning used to arrive at that statement. Alternatively, if you can prove that a contradictory statement to be true, you have proven the original statement false - therefore no proof can exist. 
