The answer to Question 2 is no. An obstruction to this being true is that $A$ can't be recovered from $\text{Rep}(A)$; we can only recover the Morita equivalence class of $A$. In particular, a monoidal product on $\text{Rep}(A)$ can be induced from a comultiplication on any algebra Morita equivalent to $A$.

For example, if $A$ is semisimple over $\mathbb{C}$ (for simplicity), then $\text{Rep}(A)$ is completely determined by $\dim Z(A)$, and in particular $A$ is Morita equivalent to $\mathbb{C}[G]$ where $G$ is any finite group with $\dim Z(A)$ conjugacy classes. Any choice of such a group gives a symmetric monoidal structure on $\text{Rep}(A)$, and these won't arise from comultiplications on $A$ in general. To be very explicit, let $A = \mathbb{C}^n$. Then any comultiplication $A \to A \otimes A$ is necessarily induced by a map of sets $[n] \times [n] \to [n]$, from which it follows that the induced tensor product has the property that a tensor product of simple representations is simple. But this isn't true for tensor products induced on representation categories of groups in general.

Monoidal products on $\text{Rep}(A)$ can be induced by even more exotic data, e.g. we could use a "comultiplication" of the form $A' \to A'' \otimes A'''$ where $A', A'', A'''$ are three different algebras Morita equivalent to $A$, or we could use an $(A, A \otimes A)$-bimodule $M$. (This exotic data still needs to satisfy suitable compatibility relations.) See also sesquialgebra.

**Edit:** Oh, and of course $A$ could be commutative and the monoidal product could be the tensor product over $A$!

For a positive result along these lines, Etingof, Nikshych, and Ostrik showed that any fusion category is the category of representations of a weak Hopf algebra (but here we don't require a symmetry).

**Edit #2:** Some thoughts on Question 1. If $V, W$ are $A$-modules then the tensor product $V \otimes W$ (the underlying field is suppressed) naturally has the structure of an $A \otimes A$-module. This gives a functor $\text{Rep}(A) \times \text{Rep}(A) \to \text{Rep}(A \otimes A)$. Let me slightly change my interpretation of "tensor product of vector spaces gives a symmetric monoidal structure" to "the monoidal structure factors as

$$\text{Rep}(A) \times \text{Rep}(A) \to \text{Rep}(A \otimes A) \xrightarrow{F} \text{Rep}(A)$$

and $F$ preserves the forgetful functor to $\text{Vect}$."

The forgetful functor is faithful and preserves colimits, so any such $F$ should also preserve colimits. Then by the Eilenberg-Watts theorem $F$ must be tensor with an $(A, A \otimes A)$-bimodule $M$. Compatibility with the forgetful functor implies that, as a right $A \otimes A$-module, $M \cong A \otimes A$, so our functor must be induced by a left $A$-module structure on $A \otimes A$. I don't think we can conclude that this comes from an algebra homomorphism $A \to A \otimes A$, but I don't know any counterexamples.