To answer these questions, the best is to note that $|\int_C D|$, the geometric realization of this total category, is equivalently the colimit of $D$, *viewed as a functor with values in the $\infty$-category of spaces*, namely, along the inclusion $Set\to Spaces$.

So

Insofar, as you believe that this colimit has significance, so do its homotopy groups. For example, if $C= \Delta^{op}$, $D$ is a simplicial set and $|\int_{\Delta^{op}} D|$ is the homotopy type of this simplicial set; if $C= BG$ for some group $G$, $D$ is a set with a $G$-action, and this homotopy type is $D_{hG}$, the homotopy orbits of $D$ under this action, etc.

The "fundamental monoid" is maybe not so relevant, in a sense because it is easily computed. First, note that it depends on more than a class in $\pi_0$ ($\pi_0$ inverts all morphisms), but really on an object in $\int_C D$. Second, given such an object $x$, it lives in a fiber over some $c\in C$, and consequently, $\hom_{\int_CD}(x,x) = \coprod_{f\in \hom_C(c,c)} \hom_{D(c)}(f_!x,x)$ where $f_! : D(c)\to D(c)$ is $D(f)$. The monoid structure is "fiberwise". But in a sense it can be easily read off of $D$, whereas understanding $|\int_C D|$ is typically more complicated.

In particular, note that the group-completion of this monoid is *not* (in general) $\pi_1(|\int_CD|, x)$, because the latter also involves "visiting" other objects than $x$. For example if $D$ is the constant functor with value a point, then $\int_C D=C$, and you'd be claiming that for an arbitrary $C$, $\pi_1(|C|,c) = $ the goup-completion of $\hom_C(c,c)$. But any group can be realized as $\pi_1(|P|)$ for a *poset* $P$ where, in particular, $\hom_P(p,p) = $ a point for any $p\in P$.

- I think the very first part of my question answers this one.