I suggest to use profunctors induced by (or simply instead of) partial functors.
First of all, a profunctor $H:A^{op}\times B\to {\rm Set}$ is said to be functorial if $H(a,-):B\to {\rm Set}$ is a representable functor for each object $a\in A$. This is equivalent to saying that $H\simeq \hom_B(F-,\, -)\,=:F_*$ for a functor $F:A\to B$.
And actually the profunctor of the colimit functor can be defined for any category, even if it lacks some colimits of the given shape: namely, consider
$$H:(C^I)^{op}\times C\to {\rm Set}\quad (\underset{I\to C}D,\,a)\mapsto
\{D\to a\text{ cocones in }C\}\,,$$
where a $D\to a$ cocone can be thought of as a natural transformation $D\to a$ to the constant functor.
Note that a diagram $D\in C^I$ has a colimit, by the very definition, if and only if $H(D,-)$ is representable.
Secondly, not only partial functors but also all categorical relations (that is, subcategories of $A\times B$), moreover any span of categories $A\overset{P}\leftarrow R\overset{Q}\to B$ do determine a profunctor in a natural way, namely,
$$P^*\otimes Q_*:A\not\to R\not\to B$$
where $P^*=\hom_A(-,\,P-);\ \ Q_*=\hom_B(Q-,\,-)$ and $\otimes$ means composition (tensor product) of profunctors, which is a colimit/coend construction that yields a quotient of $\displaystyle\bigsqcup_{r\in R}P^*(a,r)\times Q_*(r,b)$.
I guess, the $2$-category you are defining would naturally embed into the bicategory of profunctors this way.
You've basically got it. $|-|\circ F$ is perfectly well defined since you assumed $F$ is valued in simplicial sets for which $|-|$ is defined. However, note that you need $|-|\circ F$ to also be defined on morphisms in $J.$ This ties into...
No, it is not really unique. It depends on a choice of $\alpha_{S_\bullet},$ but only on that. The best way to proceed is to strengthen your construction in 1: assume that "a simplicial set has a geometric realization" means you've defined $|S_\bullet|$ along with a choice of $\alpha_{S_\bullet}.$ With that clarification, your concerns in both 1 and 2 are resolved.
Given $f:S_\bullet\to T_\bullet$ and $x\in S_\bullet,$ let $x$ be represented by $a_x:\Delta^n\to S_\bullet.$ Then we have $|f|\circ |a_x|=|f\circ a_x|.$ Since $f\circ a_x:\Delta^n\to T_\bullet$ represents the simplex $f(x),$ we see that $|f|$ sends the realization of $x,$ which is some quotient space of $|\Delta^n|$ in $|S_\bullet|,$ to the realization of $f(x).$ In other words, the geometric realization of a simplicial set map acts on simplices, now geometrically visible, in the same way as the original map did.
I think what Riehl's quote refers to is that, in general, if you have a functor $F:A\to\mathcal C$ with $A$ small and $\mathcal C$ cocomplete, you always get a cocontinuous functor $\widehat F:\widehat A\to \mathcal C$ via the co-Yoneda lemma. This is closely related to what we're doing here, except that we focused on defining geometric realization using the intended right adjoint. You can get away without this by starting with the geometric realization of maps between representable simplicial sets, which resolves the missing morphisms from (1) in a different manner.
Best Answer
Your definition of $\eta : F \Rightarrow \Delta(K)$ is correct (though you should check that it is natural—this is an easy task!)
Let $X$ be a simplicial set and let $\theta : F \Rightarrow \Delta(X)$ be another cone. Then you can define $u : K \to X$ as follows. First note that there are natural isomorphisms $K_n \cong \mathrm{Set}_{\Delta}(\Delta^n, K)$ and $X_n \cong \mathrm{Set}_{\Delta}(\Delta^n, X)$ by the Yoneda lemma; and then note that there is a map $\mathrm{Set}_{\Delta}(\Delta^n, K) \to \mathrm{Set}_{\Delta}(\Delta^n, X)$ given by $f \mapsto \theta_f$. This induces functions $u_n : K_n \to X_n$ for each $n \in \mathbb{N}$, which piece together to form a map of simplicial sets $u : K \to X$.
Finally, check that $\Delta(u) : \Delta(K) \Rightarrow \Delta(X)$ is the unique natural transformation such that $\Delta(u) \circ \eta = \theta$.