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7- I'm sorry, I misunderstood that these were your main questions. So, for the question of which structures, this amounts to answering 6-. I'll try to see later if I can answer that. For the second question however, the way everything is set up makes it very easy to answer : a good point in $\mathcal M$ is an ultrafunctor $*\to \mathcal M$. This is clearly stable under composition of ultrafunctors : if $F:\mathcal{M\to N}$ is an ultrafunctor, so is $*\to \mathcal M\to\mathcal N$, and in particular, $F$ preserves "nice" objects. A corollary of how everything is set up is also (for free) that you get a morphism of ultramonoids (in the sense I described, so a morphism of compact Hausdorff topological monoids) $End(M)\to End(F(M))$.

It looks like this map is always injective by definition of $\mathcal U\wr \mathcal V_\bullet$, but most likely almost never surjective. For instance take $K =*$, then this is $\mathcal M\to (\prod_J \mathcal M)/\mathcal U$, which is almost never surjective.

7- I'm sorry, I misunderstood that these were your main questions. So, for the question of which structures, this amounts to answering 6-. The answer seems to be "almost never" - my guess would be "if and only if $M=*$". It makes sense to a certain extent : what would be the compact Hausdorff topology on the set of elementary embeddings/morphisms ?

For the second question however, the way everything is set up makes it very easy to answer : a good point in $\mathcal M$ is an ultrafunctor $*\to \mathcal M$. This is clearly stable under composition of ultrafunctors : if $F:\mathcal{M\to N}$ is an ultrafunctor, so is $*\to \mathcal M\to\mathcal N$, and in particular, $F$ preserves "nice" objects. A corollary of how everything is set up is also (for free) that you get a morphism of ultramonoids (in the sense I described, so a morphism of compact Hausdorff topological monoids) $End(M)\to End(F(M))$.

6- I'm not sure what the nice objects are in the case of "models of a first order theory". Essentially these are the models $M$ such that for any two sets $J,K$, function $f: J\to \beta K$ and ultrafilter $\mathcal U$ on $J$, the canonical morphism $\prod_J ((\prod_K M)/\mathcal f(j))/\mathcal U \to (\prod_K M)/\mathcal V$ is an isomorphism, where $\mathcal V = \lim_\mathcal U f(j)$ in $\beta K$.

6- I'm not sure what the nice objects are in the case of "models of a first order theory". Essentially these are the models $M$ such that for any two sets $J,K$, function $f: J\to \beta K$ and ultrafilter $\mathcal U$ on $J$, the canonical morphism $(\prod_K M)/\mathcal V \to \prod_J ((\prod_K M)/\mathcal f(j))/\mathcal U$ is an isomorphism, where $\mathcal V = \lim_\mathcal U f(j)$ in $\beta K$, equivalently $\mathcal U\wr \mathcal V_\bullet$ where $\mathcal V_\bullet = f$.

EDIT to answer the questions in a comment below (rather than as a long sequence of further comments):

1- I used "pointed category" to mean something more naive than in the nLab, namely a category $C$ with a point $*\to C$. Morphisms of such are functors $f : C\to D$ that preserve the point, and $2$-morphisms are natural transformations that are the identity on the point (this is an unfortunate clash of terminology: one can define a "pointed object" in an arbitrary category, and "pointed objects of $Cat$" do not coincide with the nLab's pointed categories). With this definition, the (usual) $1$-category of monoids is equivalent to the $(2,1)$-category of pointed categories for which the point $*\to C$ is essentially surjective.

2- "which question": I was adressing the question about ultramonoids (at first I thought also posets, but in the end, no). I don't know if I'm using the same definition of ultramonoid as you are, because it's unclear to me what "an ultracategory with one object" means in the same way that it's unclear to me what "a category with one object" means.

If you mean in the strictest sense and using Definition 7, then no I am not using the same definition (a choice which my first paragraph tries to explain). There are 2 reasons I did not use the same definition: a- what I explained in the first paragraph of this (partial) answer and b- the strict definition is not invariant under the several presentations of "ultracategories" and not invariant under equivalence of ultracategories. I know that this means I'm not technically answering your question (in any case, I wasn't answering all of it, this was just a contribution ! I'm sorry if I made it sound like it was supposed to be everythin). But maybe in the situation you're interested in, the definition I used might be more relevant ?

3- The notation $*_{\beta I}\to \mathcal M_{\beta I}$ refers to $*^I\to \mathcal M^I$ in the language of definition 7, specifically the map that picks out this one object. Sorry again, Definition 1 was clearer to me, which is why I phrased my answer in this language (note that the two definitions are equivalent in a suitable sense).

4- I define the $(2,1)$-category of ultramonoids as a full subcategory of the category of pointed ultracategories (in the sense I described earlier). In particular, morphism categories (rather than morphism sets) are categories of ultrafunctors and, I guess but am not sure about the terminology, ultra natural transformations.

5- A "point" of a category is an object therein, equivalently a functor $*\to C$. I used this latter perspective (which may seem unnecessarily pedantic) to define a point of an ultracategory, as an ultrafunctor from the trivial ultracategory $*$ to $\mathcal M$. That's the content of the paragraph following "The first question is..." - I was trying to identify those more concretely.

6- I'm not sure what the nice objects are in the case of "models of a first order theory". Essentially these are the models $M$ such that for any two sets $J,K$, function $f: J\to \beta K$ and ultrafilter $\mathcal U$ on $J$, the canonical morphism $\prod_J ((\prod_K M)/\mathcal f(j))/\mathcal U \to (\prod_K M)/\mathcal V$ is an isomorphism, where $\mathcal V = \lim_\mathcal U f(j)$ in $\beta K$.

7- I'm sorry, I misunderstood that these were your main questions. So, for the question of which structures, this amounts to answering 6-. I'll try to see later if I can answer that. For the second question however, the way everything is set up makes it very easy to answer : a good point in $\mathcal M$ is an ultrafunctor $*\to \mathcal M$. This is clearly stable under composition of ultrafunctors : if $F:\mathcal{M\to N}$ is an ultrafunctor, so is $*\to \mathcal M\to\mathcal N$, and in particular, $F$ preserves "nice" objects. A corollary of how everything is set up is also (for free) that you get a morphism of ultramonoids (in the sense I described, so a morphism of compact Hausdorff topological monoids) $End(M)\to End(F(M))$.

8- By strongly connected I meant "only one isomorphism class". I'm not sure there is a standard terminology and what it is, if there is.