Why is entropy generation zero at infinitesimal temperature gradient? First question: Is the question I asked valid in the first place? Is it really true? 
Second question: If so, is it because the exchanged heat between the bodies between which the temperature gradient exists is also infinitesimal and hence negligible? The explanation I provided is not convincing for me as, in physics, "infinitesimal" is not always synonymous with "negligible". 
 A: In the Transport Phenomena by Bird et al reference I provided in my answer to your previous question, they show that the rate of entropy generation per unit volume resulting from temperature gradients is equal to the thermal conductivity times the square of the temperature gradient divided by the (absolute) temperature.  So, if you integrate this over the entire volume of gas or fluid, you get the global rate of entropy generation within the system.  You can then judge for yourself quantitatively whether this is negligible for the specific situation you are considering.
A: Chester Miller's answer is correct as it stands. I am just filling in some details, if you are looking for the mathematical consequences of introducing an infinitesimal temperature gradient.
As Chester Miller pointed out, total entropy generated within a volume $V$ per unit time is $$\dot{S}=\int_VdV~\frac{k}{T}|\nabla T|^2$$ in which $k$ is thermal conductivity and $T$ is temperature. Now if everywhere $|\nabla T|\sim\epsilon$, an arbitrarily small quantity, then $\dot{S}\sim\epsilon^2$. This means that to linear order in $\epsilon$ there is no entropy generation, and the correction comes in only at second order in $\epsilon$. In other words, a Taylor expansion of $\dot{S}$ in $\epsilon$ begins directly with the $\epsilon^2$ term.
To answer your question: Infinitesimal temperature gradient causes zero entropy generation, with an error of second or higher order in the infinitesimal quantity. There is no straightforward yes 
/no answer; mathematically speaking, whether you are going to assume that entropy generation is zero due to an infinitesimal temperature gradient depends on the level of approximation you wish to employ.
