Let me add some additional remarks on the enumeration question:

*How many such spaces $A$ are there sitting over $B\times C$ such that the homotopy pushout
$$
B \leftarrow A \to C
$$
is contractible?*

If $B\cup_A C$ is contractible, then the sum of the maps $\Sigma A \to \Sigma B$ and $\Sigma A \to \Sigma C$ gives a homology isomorphism
$$
\Sigma A \to \Sigma B \vee \Sigma C\, .
$$
If $A$ is connected, then we conclude that the map is also a weak homotopy equivalence, so $\Sigma A$ is a wedge of $\Sigma B$ and $\Sigma C$ in this case. The converse to this statement is also true.

Hence, we are reduced to classifying those spaces $A$ whose suspension splits as a wedge of $\Sigma B$ and $\Sigma C$. Hence, this problem can be formulated as a kind desuspension problem.

There is a classifying space ${\cal D}(B,C)$ which is the realization of (the nerve of) a category whose objects are spaces $A$ with structure map $A \to B\times C$ such that $\text{hocolim}(B \leftarrow A \to C)$ is contractible. A morphism of this category is a weak homotopy equivalence of spaces over $B\times C$.

Classification Result: Assume $A,B,C$ are $1$-connected. There is a function
$$
\pi_0({\cal D}(B,C)) \to \{B\vee C, B\wedge C\}
$$
(the target is the abelian group of homotopy classes of stable maps) which is a bijection in a metastable range
$$
\max(b,c) \le 3\min(r,s)-1\, ,
$$
where $b$ is the homotopy dimension of $B$ (the CW complex of minimal dimension having the homotopy type of $B$), $c$ is the homotopy dimension of $C$, $r$ is the connectivity of $B$ and $s$ is the connectivity of $C$.
Here I have given $B$ and $C$ basepoints.

The function in (4) is easily described as the stable homotopy class of the weak map
$$
\Sigma B\vee \Sigma C \overset{\simeq}\leftarrow \Sigma A \to \Sigma (B \wedge C)
$$
where the right arrow is given by suspending the composition
$A \to B\times C \to B\wedge C$.

If we want to work beyond the metastable range, the homotopy type of ${\cal D}(B,C)$ can be determined "up to extensions" via the coefficients of the identity functor in homotopy functor calculus.

See the paper

Klein, John R.; Peter, John W. Fake wedges. Trans. Amer. Math. Soc. 366 (2014), no. 7, 3771–3786

which extends my earlier paper:

John R. Klein, Structure theorems for homotopy pushouts. I. Contractible pushouts, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 2, 301–324.

Structure theorems for homotopy pushouts. I. Contractible pushouts.by John Klein (MR1490203) is all about contractible pushouts. $\endgroup$ – Jeff Strom Jan 22 at 17:08