Why can't some mathematical statement (or whatever is the correct term) be both true and false? ...[H]ow can we be sure that no one will ever find a counter example?
We can't be sure. Logics are created by humans, and humans are not gods. Similar to any other scientific statement, we can show existence, but not universality. The existence of a mistake can be shown, but the universality of a logic never having a mistake will never be demonstrable, no matter what trickery is employed. It is a modern schtick to pretend that this issue can be bypassed with "model theory" (which is certainly a useful subject, and is a common context of Gödel's Theorems), but it is just pretend.
There are some things we can do to give us confidence though. We can reduce the complexity of axioma to the point where the assumptions are "obvious". Sometimes even obvious assumptions have problems and need to be improved though (such as unrestricted set comprehension, which led to Russell's Paradox).
Confidence can also be improved by a logic being used by many different people with many different purposes, without a problem ever being discovered. But it will never be 100% confidence. Eventually, actually finding an inconsistency in the logic would be more interesting than what you are working on anyway, so there is no need to worry. It's like worrying about finding something amazing.
We can, however, assert with 100% confidence that a logical system $A$ is no less correct than another system $B$ by encoding $A$ into $B$ and proving the consistency of $A$ with the assumptions of $B$.
Then I have heard about Gödel's incompleteness theorems, second of which says (at least this is how I have interpreted it) that an axiomatic system cannot prove its own consistency.
This is not an accurate interpretation of the theorem. It is easy to construct a logic that proves it's own consistency, just construct an inconsistent logic with vacuous implication. From an inconsistency follows the assertion of false, and from false follows anything, including consistency.
A better characterization of Gödel's Second Incompleteness Theorem (but still nowhere nearly as formal as the actual theorem) is:
- A sufficiently interesting logic that can prove it's own consistency is inconsistent
So doesn't Gödel's second incompleteness theorem say basically that "anything is possible"?
If your logic is sufficiently interesting and contains a proof of its own consistency, then yes. Otherwise and more commonly, no.