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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).$$ I do not think it follows "by definition" that J^* is subcanonical, but maybe I am missing something. 1 This not a complete answer but too long for a comment: First a quick remark: 1.) and 4.) generate the same topology. Now let geometric realization be denoted by G. Consider the geometric morphism $$\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),$$ where y denotes the Yoneda embedding. Since all of the Grothendieck topologies you mentioned are extensive, it follows that that the J^* sieves are the ones generated by the ones of the form $$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.