regarding the infinite cross section in Rutherford scattering

The differential cross section of Rutherford scattering blows as $\theta\rightarrow 0$. People express this fact as "The Coulombic potential is of long range". But I am seeing the opposite: $\theta \rightarrow 0$ is when the incident particle is not affected by the potential, and therefore a blow when $\theta \rightarrow 0$ means short range potential.

That was first. Secondly, I cannot see how the differential cross section is related to (some kind) of probabilities when infinities are involved.