I think that the answer to (1) is likely no in the general case where we consider finite dimensional $A$-modules with an infinite-dimensional $A$. In this context, Allen's suggestion can't work simply because $A$ doesn't live in $\operatorname{Rep}(A)$!

The reason is that we typically can't reconstruct an infinite-dimensional algebra from its finite dimensional representations. For instance, the algebra $A = k[x_1, x_2, \ldots] / (x_1^2 = 0, x_{i + 1}^2 = x_i)$ has no nontrivial finite dimensional representations whatsoever! Of course, there is the usual symmetric monoidal structure on $\operatorname{Rep} A = \operatorname{Vec}$, so any coalgebra structure on $A$ will work in this case (but are there any?).

Something like Allen's suggestion should work, for purely formal reasons, for either finite-dimensional $A$ or if we consider the larger category of all $A$-modules. The dual of your question (where we ask for an algebra structure on a coalgebra inducing a given symmetric monoidal structure on the category of coalgebras) is more or less the classical theory of Tannaka duality, which gives an affirmative answer.

I think that the answer to (1) is likely no in the general case where we consider finite-dimensional $A$-modules with an infinite-dimensional $A$. In this context, Allen's suggestion can't work simply because $A$ doesn't live in $\operatorname{Rep}(A)$!

The reason is that we typically can't reconstruct an infinite-dimensional algebra from its finite-dimensional representations. For instance, the algebra $A = k[x_1, x_2, \ldots] / (x_1^2 = 0, x_{i + 1}^2 = x_i)$ has no nontrivial finite-dimensional representations whatsoever! Of course, there is the usual symmetric monoidal structure on $\operatorname{Rep} A = \operatorname{Vec}$, so any bialgebra structure on $A$ will work in this case (but are there any?).

Something like Allen's suggestion should work, for purely formal reasons, for either finite-dimensional $A$ or if we consider the larger category of all $A$-modules. The dual of your question (where we ask for an algebra structure on a coalgebra inducing a given symmetric monoidal structure on the category of coalgebras) is more or less the classical theory of Tannaka duality, which gives an affirmative answer.

EDIT: I fixed the algebra $A$ to make it actually have no finite-dimensional representations. Unfortunately, this algebra does admit a bialgebra structure. I'll try to think of an example of an $A$ that has $\operatorname{Vec}$ as its category of finite-dimensional representations but provably admits no bialgebra structure.

I think that the answer to (1) is likely no in the general case where we consider finite dimensional $A$-modules with an infinite-dimensional $A$. In this context, Allen's suggestion can't work simply because $A$ doesn't live in $\operatorname{Rep}(A)$!

The reason is that we typically can't reconstruct an infinite-dimensional algebra from its finite dimensional representations. For instance, the algebra $A = k[x_1, x_2, \ldots] / (x_{i + 1}^2 = x_i)$ has no nontrivial finite dimensional representations whatsoever! Of course, there is the usual symmetric monoidal structure on $\operatorname{Rep} A = \operatorname{Vec}$, so any coalgebra structure on $A$ will work in this case (but are there any?).

Something like Allen's suggestion should work, for purely formal reasons, for either finite-dimensional $A$ or if we consider the larger category of all $A$-modules. The dual of your question (where we ask for an algebra structure on a coalgebra inducing a given symmetric monoidal structure on the category of coalgebras) is more or less the classical theory of Tannaka duality, which gives an affirmative answer.

I think that the answer to (1) is likely no in the general case where we consider finite dimensional $A$-modules with an infinite-dimensional $A$. In this context, Allen's suggestion can't work simply because $A$ doesn't live in $\operatorname{Rep}(A)$!

The reason is that we typically can't reconstruct an infinite-dimensional algebra from its finite dimensional representations. For instance, the algebra $A = k[x_1, x_2, \ldots] / (x_1^2 = 0, x_{i + 1}^2 = x_i)$ has no nontrivial finite dimensional representations whatsoever! Of course, there is the usual symmetric monoidal structure on $\operatorname{Rep} A = \operatorname{Vec}$, so any coalgebra structure on $A$ will work in this case (but are there any?).

Something like Allen's suggestion should work, for purely formal reasons, for either finite-dimensional $A$ or if we consider the larger category of all $A$-modules. The dual of your question (where we ask for an algebra structure on a coalgebra inducing a given symmetric monoidal structure on the category of coalgebras) is more or less the classical theory of Tannaka duality, which gives an affirmative answer.