But nowhere does he proves that all derivatives of f(x) are not zero for x=0 and x=pi. f(x) clearly is zero there. So saying that F(pi) + F(0) is an integer proves nothing if both are equal to zero...?

You don't actually need F(pi) + F(0) to be non-zero. The proof shows that the integral is positive, but arbitrarily small. The means that F(pi)+F(0) must be positive, but arbitrarily small. If F(pi)+F(0) is a positive integer, it fails the arbitrarily small test. If F(pi)+F(0) is a non-positive integer (including 0), it fails the positive test.