$\int_a^af(x) \, dx$ always $0$?

You are right, when talking about the Riemann Integral, then $$ \int_a^a f(x)\; dx $$ always equal to zero. If this integral is defined, then it is zero. To see this, you just have to write out the definition of the integral: $$ \int_a^a f(x)\; dx = \lim_{n\to \infty}\sum_{i=0}^{n} f(x_i^*) \, \Delta x $$ Here $\Delta x = \frac{a - a}{n} = 0$, so all the sums will be zero, and so the limit will be zero.

No, for $f(x)$ any continuous function, $\int_a^a f(x)dx= 0$. As for non-continuous functions, my first thought was a "Dirac delta function" for which $\int_C \delta(x)dx= 1$ for any set, C, containing 0. However, a "delta function" is not a true function- it is a "generalized function" or "distribution".