This isn't a full answer, but too long for a comment. I suppose it comes closest to trying to answer Question 3 or the general question of whether the hypercube design can be improved.
**Definition** Given a hypergraph $G=(\{v_1, \dots, v_n\}, E)$, the *dual* of $G$ is the hypergraph $H$ with $V(H)=E(G)$ and $E(H)=\{\{e\in E(G): v_i\in e\}: i\in [k]\}$ (in other words, each edge of $H$ is a maximal collection of edges from $G$ which are incident with a single vertex).
Let $H_{n,k}$ be the dual of $K_n^{k}$, the complete $k$-regular hypergraph on $n$ vertices. Note that the dual of $H_{n,k}$ is isomorphic to $K_n^k$.
(It seems to me that this hypergraph must have been studied before, but I couldn't find any reference to it. One possible lead is that $H_{4,2}$ is what you call the *complete quadrilateral*.)
**Claim 1.**
$H_{n,k}$ is a $\binom{n-1}{k-1}$-uniform $k$-regular hypergraph with $\binom{n}{k}$ vertices and $n$ edges.
*Proof.* In $K_n^k$, every vertex is incident with $\binom{n-1}{k-1}$ edges, every edge has order $k$, there are $\binom{n}{k}$ edges, and $n$ vertices.$\square$
**Claim 2.** $H_{n,k}$ is a pooling design.
*Proof.*
Every vertex in $H_{n,k}$ is incident with $k$ edges, so $|x^*|=k$. If $X$ is a set of vertices with $|X|>1$ (which corresponds to a set of more than one edge in $K_n^k$, which spans more than $k$ vertices in $K_n^k$) then $|X^*|>k$. So $x^*\neq X^*$ if $|X|>1$.$\square$
The compression rate of $H_{n,k}$ is $\frac{n}{\binom{n}{k}}$ which is minimized when $k=\lfloor{n/2}\rfloor$. Also note that ratio of the uniformity to the number of vertices is $\binom{n-1}{k-1}/\binom{n}{k}=k/n$. So there is a tradeoff
when minimizing the compression rate, since the uniformity and degree increase when we increase $k$.
Some more examples: $H_{5,2}$ is 4-uniform with 10 vertices and 5 edges giving a compression ratio of $1/2$. $H_{6,3}$ is 10-uniform with 20 vertices and 6 edges, giving a compression ratio of $3/10$. $H_{7,3}$ is 15-uniform with 35 vertices and 7 edges, giving a compression ratio of $1/5$. Note that the hypercube design with $D=3$ is 9-regular with 27 vertices and 9 edges and thus a compression ratio of 1/3, so $H_{6,3}$ and $H_{7,3}$ compare favorably in this case.
**Update 1**. (It seems best to update my previous answer rather than write a new one.)
After thinking it over some more, I think I have an alternate characterization of pooling designs which both makes it easier to check if $H$ is a pooling design and elucidates some features of pooling designs. In particular, this gives a simple proof of the Propositions in your answer.
**Claim 3** $H$ is a pooling design if and only if $x^*\not\subseteq y^*$ for all distinct $x,y\in V(H)$.
*Proof.*
($\Rightarrow$) Suppose there exists distinct $x,y\in V(H)$ such that $x^*\subseteq y^*$. Then $y^*=\{x,y\}^*$ and thus $H$ is not a pooling design.
($\Leftarrow$) Suppose $H$ is not a pooling design; that is, suppose there exists $y\in V(H)$ and $Y\subseteq V(H)$ with $Y\neq \{y\}$ such that $y^*=Y^*$. Since $Y\neq \{y\}$, there exists $x\in Y$ such that $x\neq y$. Since $x\in Y$, we have $x^*\subseteq Y^*=y^*$.
$\square$
**Corollary 1**
Let $H$ be a hypergraph and let $G$ be the dual of $H$.
$H$ is a pooling design if and only if $e\not\subseteq f$ for all distinct $e,f\in E(G)$.
*Proof.*
($\Rightarrow$) Suppose $H$ is a pooling design. Choose distinct $e,f\in E(G)$ which correspond to distinct $x, y\in V(H)$ respectively. Since $x^*\not\subseteq y^*$, we have $e\not\subseteq f$.
($\Leftarrow$) Suppose $e\not\subseteq f$ for all distinct $e,f\in E(G)$. Choose distinct $x,y\in V(H)$ which correspond to distinct $e,f\in E(G)$. Since $e\not\subseteq f$, we have $x^*\not\subseteq y^*$.
$\square$
**Corollary 2**
Let $H$ be a hypergraph with $e$ edges and $n$ vertices such that $\binom{e}{\lfloor{e/2}\rfloor}<n$. Then $H$ is not a pooling design.
*Proof.*
Let $G$ be the dual of $H$ and note that $G$ has $e$ vertices and $n$ edges. Since $|E(G)|=n>\binom{e}{\lfloor{e/2}\rfloor}=\binom{|V(G)|}{\lfloor{|V(G)|/2}\rfloor}$, [Sperner's theorem][1] implies that there exists distinct $e,f\in E(G)$ such that $e\subseteq f$. Thus $H$ is not a pooling design by Corollary 1.
$\square$
In particular, this proves that every pooling design on $4\leq n\leq 6$ vertices has at least 4 edges, every pooling design on $7\leq n\leq 10$ vertices has at least 5 edges, etc.
**Update 2**.
Again, after considering some more, I now think it's clearer to just remain in the setting of the hypergraph $G$ and forget about taking the dual.
For example, let's compare the $K_8$-design to the hypercube design with $D=3$. In the $K_8$-design, each edge is a sample (there are 28), each vertex is a test pooling the samples which are incident with that vertex (there are 8), each test pools 7 samples (since the degree of each vertex is 7), and each sample will be used twice (since $K_8$ is 2-uniform). As I mentioned in a comment, this is better than the hypercube design in every parameter. Also you can see that if exactly one sample is infected, say the edge $\{i,j\}$, then exactly two tests (test $i$ and test $j$) will come back positive.
For another example, let's compare the $K_{13}$-design to the hypercube design with $D=4$. The $D=4$ hypercube design handles 81 samples using 12 tests each of which has size 27 and each sample is used 4 times. The $K_{13}$-design handles 78 samples using 13 tests, but each test has size 12 and each sample is only used 2 times.
For a final example, let's compare the $K_{9,9}$-design (that is, a complete bipartite graph with 9 vertices in each part) to the $D=4$ hypercube design. The $K_{9,9}$-design handles 81 samples using 18 tests, each of which has size 9 and each sample is used 2 times; however, this design has the additional feature that if three tests come back positive, then we will know exactly which two samples are infected. Neither the $K_{13}$-design, nor the $D=4$ hypercube design have that property.
[1]: https://en.wikipedia.org/wiki/Sperner%27s_theorem