Why not both true and false? Why can't some mathematical statement (or whatever is the correct term) be both true and false?

For example we can prove (e.g. by induction) that $1+2+3+\cdots+n=\frac{n(n+1)}{2}$ for all positive integers $n$. But how can we be sure that no one will ever find a counter example? What if someone claims that $1+2+3+\cdots+1000$ equals (e.g.) 500567 and not 500500, which is what the above formula claims.

Another example: Why is it impossible for someone to come up with three integer $a$, $b$ and $c$, for which $a^3+b^3=c^3$ (contradicting Fermat's Last Theorem)? This bothers me even in the simple intuitive level.

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. So doesn't Gödel's second incompleteness theorem say basically that "anything is possible"? ...that there can be an integer $n$ for which $1+2+3+\cdots+n \neq \frac{n(n+1)}{2}$ or that there can be integers $a$, $b$ and $c$ for which $a^3+b^3=c^3$?
Gödel's theorem could be more accurately interpreted as saying that we can never be sure of the consistency of a sufficiently complex system. We can't be sure, for instance, that the Peano Axioms don't prove $1+1=3$. We sure hope this isn't the case, but no proof would convince us otherwise (and it's probably not, since the Peano Axioms have an intuitive model as being the natural numbers with addition and multiplication).

However, it's still true that $1+1=2$ even if the Peano Axioms say otherwise (indeed, if they proved $1+1=3$, they would also have to prove $1+1\neq 3$, and also every other statement you could possibly make within that system). In fact, we can say that, if a (suitably complex) system is inconsistent, then it admits both a proof and a disproof of every statement - this is the principle of explosion.

The difference is that there is an intended model of the Peano Axioms - the natural numbers with addition and multiplication. This is clearly well-defined and certain things are undeniably true of them. We would therefore expect that the Peano Axioms are, in fact, consistent (though we can't prove it) - and, if it is consistent, everything it proves is true and undeniably so. Even if PA were inconsistent, we would still expect proofs like the $1+2+\ldots+n=\frac{n(n+1)}2$ to be work since they are leveraging such simple properties of the structure of the natural numbers.

The point here is that "truth" and "proof" are distinct statements - but we tend to identify them because we assume our logical systems are consistent, or at least assume that the bits of them we actually use are consistent.
We would have hoped that in mathematics every reasonable statement is either true or false. As Gödel showed, that is not the case: In any mathematical system, there are two possibilities: Either most statements are either true or false but some are neither true or false. Or the system is contradictory, which means every statement is both true and false at the same time which means the whole system is useless. 

Now you asked "Is it possible that a simple statement is both true and false"? We don't know which of the two kinds of systems we are using. We very very much hope that it is not contradictory. In that case, no, a simple statement can be either true, or false, or neither at all. If our system is contradictory, then yes, that simple statement will be both true and false. Actually, every statement will be both true and false. 

In your example, suppose we have a proof that a certain sum is 500,500 but we also calculated correctly that the same sum is 500,567. Then every mathematical statement is both true and false: We have just shown that 500,500 = 500,567, therefore $0 = 17$, therefore $0 = 1$. Then as an example, for every $a, b, c &gt; 0$ and $n \ge 3$,

$a^n + b^n = 1 \cdot (a^n + b^n - c^n) + c^n = 0 \cdot (a^n + b^n - c^n) + c^n = c^n$

So every possible tuple $(a, b, c, n)$ is a counterexample to Fermat's Last Theorem!