I don't really see a way to give a single answer here, these are 7 different questions (well actually a lot more than 7 if we count all the different flavours of cubes... so more like 30 differents questions).

Though, a common pattern I think is going to hold to answer any of these question (and I'll give the answer in one case to illustrate it), is that the first step is to understand what are colimits in the small category. In the case of $\Delta$, this is done by the following (from the paper of Calin Tataru you cite)

Prop: The embedding of $\Delta \to \text{Pos}$ preserves colimits.

Indeed as it is a fully faithful embedding this completely characterizes colimits in $\Delta$: Given any diagram in $\Delta$, we can compute its colimits in $\text{Pos}$, if the colimit is in $\Delta$ then it is a colimit in $\Delta$, otherwise there is no colimit in $\Delta$. Once you know what are the colimits in the small category, you can think of it as a colimit sketch, and the completion is the category of models of that sketch.

To put it another way, it is generally easy to understand the free cocompletion that preserves an explicitely given set of colimits. What is hard with these conservative cocompletion is that the class of colimits isn't explicitly given, so the first step is to figure out what are the colimits that need to be preserved.

Let's do the example of the category of $G$-Orbit, because it is fairly straightforward how to do it. I claim that:

Prop: A diagram in the category of $G$-Orbits has a colimit if and only if it is connected.

Proof: This is fairly easy to check that a colimit of a connected diagram of $G$-orbits taken in the category of $G$-sets is again a $G$-orbit. Conversely, it is also easy to check that coproducts and initial objects never exists in the category of $G$-orbits.

It then follows that the conservative cocompletion of $G$-Orbits is the free cocompletion that presverse all these connected colimits, so it is actually just the completion under coproduct, in particular we get:

Theorem: The conservative cocompletion of the category of $G$-orbits is the category of $G$-set.

I feel like using well behaved embeddings of the small category onto a cocomplete category is a good way to start studying the colimits in the small category (like the embedding $\Delta \to \text{Pos}$ in the paper you cite, or the embedding $G$-Orbit to $G$-set I'm using here to understand colimits) the fact that these embedding end-up being the cocompletion is I think not strictly neccessary.

I don't really see a way to give a single answer here, these are 7 different questions (well actually a lot more than 7 if we count all the different flavours of cubes, and of the other ones, probably more like 30 differents questions).

Though, a common pattern I think is going to hold to answer any of these question (and I'll give the answer in one case to illustrate it), is that the first step is to understand what are colimits in the small category. In the case of $\Delta$, this is done by the following (from the paper of Calin Tataru you cite)

Prop: The embedding of $\Delta \to \text{Pos}$ preserves colimits.

Indeed as it is a fully faithful embedding this completely characterizes colimits in $\Delta$: Given any diagram in $\Delta$, we can compute its colimits in $\text{Pos}$, if the colimit is in $\Delta$ then it is a colimit in $\Delta$, otherwise there is no colimit in $\Delta$. Once you know what are the colimits in the small category, you can think of it as a colimit sketch, and the completion is the category of models of that sketch.

To put it another way, it is generally easy to understand the free cocompletion that preserves an explicitely given set of colimits. What is hard with these conservative cocompletion is that the class of colimits isn't explicitly given, so the first step is to figure out what are the colimits that need to be preserved.

Let's do the example of the category of $G$-Orbit, because it is fairly straightforward how to do it. I claim that:

Prop: A diagram in the category of $G$-Orbits has a colimit if and only if it is connected.

Proof: This is fairly easy to check that a colimit of a connected diagram of $G$-orbits taken in the category of $G$-sets is again a $G$-orbit. Conversely, it is also easy to check that coproducts and initial objects never exists in the category of $G$-orbits.

It then follows that the conservative cocompletion of $G$-Orbits is the free cocompletion that presverse all these connected colimits, so it is actually just the completion under coproduct, in particular we get:

Theorem: The conservative cocompletion of the category of $G$-orbits is the category of $G$-set.

I feel like using well behaved embeddings of the small category onto a cocomplete category is a good way to start studying the colimits in the small category (like the embedding $\Delta \to \text{Pos}$ in the paper you cite, or the embedding $G$-Orbit to $G$-set I'm using here to understand colimits) the fact that these embedding end-up being the cocompletion is I think not strictly neccessary.