This not a complete answer but too long for a comment:
First a quick remark: 1.) and 4.) generate the same topology.
Now let geometric realization be denoted by $G$. Consider the geometric morphism $$\bar G=\left(G^*,G_*\right):Psh\left(Top\right) \to Psh\left(sSet\right),$$ and the geometric morphism $$J=\left(a,i\right):Sh_J\left(Top\right) \to Psh\left(Top\right)$$ corresponding to the Grothendieck topology $J$ (I have used $J$ to denote the geometric embedding since there is a one-to-one correspondence between geometric embeddings and Grothendieck topologies).
The topos of sheaves on $sSet$ with respect to what you call $J^*,$ is the pullback topos $$Sh_{J^*}\left(sSet\right):=Psh\left(sSet\right) \times_{Psh\left(Top\right)} Sh_J\left(Top\right).$$
Let $S:Top \to sSet$ denote the singular nerve functor. Note that $G^*$ is the left-Kan extension along Yoneda of the functor $$y_G:Top \to Psh\left(sSet\right),$$
$$y_G\left(T\right):X \mapsto Hom_{Top}\left(G\left(X\right),T\right) \cong Hom_{sSet}\left(X,S\left(T\right)\right),$$
i.e. for topological space $T,$ $$G^*\left(T\right)=y\left(S\left(T\right)\right),$$$G^*\left(y\left(T\right)\right)=y\left(S\left(T\right)\right),$$
where $y$ denotes the Yoneda embedding.
Since all of the Grothendieck topologies you mentioned are extensive, it follows that that the $J^*$ sieves are the ones generated by the ones of the form
$$S\left(\underset{j} \coprod V_j\right) \to S\left(V\right).$$
I do not think it follows "by definition" that $J^*$ is subcanonical, but maybe I am missing something.
