Consider statements of the form $\forall x\in X:P(x)$. Is it possible that such a statement is proven to be unprovable?
I think not, and here is my argument: if we proved that the statement is unprovable, we would also prove that we will never be able to find a counter-example (an $x$ for which $\neg P(x)$, because that would disprove the statement), therefore proving the statement.
Is my argument correct?
Is it possible, that the statement is unprovable, even though we cannot prove its unprovability?
The logic of your informal argument is not right. If $\forall\,x\in X \cdot P(x)$ is unprovable, it does not follow that $\lnot P(x)$ is unsatisfiable. $\forall\,x\in X \cdot P(x)$ could be unprovable because it is false, e.g., $\forall\,x\cdot x \not= x$. It is true in an informal sense that a proof that a universal sentence is not disprovable constitutes a proof that it is true.
As for your last question: it needs to be asked in a more rigorous setting. At an informal level, there are all sorts of things that we can't prove. Boolos's excellent book The Logic of Provability provides a rigorous account of the logic of provability for theories of arithmetic.