Question: Based on my understanding, the differential cross section $\frac{d\sigma}{d\Omega}$ gives us the differential area of incident particles that projects onto a differential piece of solid angle on the detector. When one calculates total cross section, one integrates over the entire solid angle; i.e. $\sigma = \int d\sigma = \int \frac{d\sigma}{d\Omega}d\Omega$. I do not comprehend this. The total cross section is indicative of the likelihood of an interaction between the incident beam and the target, and let us assume that the incident beam is a beam ranging across all impact parameters. If at impact parameter large enough the incident particles do not deflect and thus has a scattering angle of close to zero, wouldn't there be an unbounded range of impact parameter scattering onto near zero angles, and thus any integration that passes through zero will result in a blow up? (not just for rutherford scattering)

Answer: 
I think any cross section is just a number, which hints at how more/less number of particles (electrons/protons etc) can be ejected/absorbed(interact) with the application of an external field. So are the terminologies like absorption cross section, photoionization cross section etc.
I agree with Sergei that the definition of the differential cross-section is legitimate as theta goes to zero. However, I would like to interpret the convergent case when theta=0.

I come up with two ways to understand this:

1. Consider the famous small hole diffraction giving rise to the airy disc pattern. If we invert the "opacity" of the diffracting plane, i.e. make it a small disc-like obstacle standing in the middle of incoming plane wave, then we get the famous Arago spot. But in the very far field, there should be an Arago spot + intensity coming from the major part of the unblocked wave (which is a delta function by Babinet's theorem). Hence the amplitude of the far field at theta=0 should be some complex combination of a delta (infinite density from original plane wave) and also an Arago spot (relocation of supposedly blocked part into edge diffraction according to wave mechanics). Therefore, it is a rather "legitimate" thing to argue that intensity at theta=0 is meaningful.
2. The Born approximation is built upon Fermi's Golden rule. From the point of view of time-dependent perturbation, we can rather interpret the theta=0 amplitude as the first order correction of the coefficient of the initial state. For non-convergent case, it is obvious that higher order of correction is needed.