Partial answer to (1).
As far as I'm aware, the only definition of a finite-dimensional semisimple algebra over a linear operad is provided by Etingof. Here is an attempt to generalize it.
Let $O$ be a symmetric operad in $(C, \otimes, 1)$, an abelian closed symmetric monoidal category, and let $A$ be an $O$-algebra in $C$, with structure maps $\alpha_n: O(n) \to [A^{\otimes n}, A]$. Define $A$-modules in $C$ the usual way; the 0 object naturally inherits a trivial $A$-module structure.
We can define families of endomorphisms by currying (or internal hom-tensor adjunction) $[A^{\otimes n}, A] \cong [A, [A^{\otimes n-1}, A]]$. These are $\alpha_n(o)(-)(a_1 \otimes \dots \otimes a_{n-1}): A \to A,$ for fixed parameters. The collection of all such endomorphisms is a (possibly non-unital) subring of the endomorphism ring $C(A,A)$, we will denote this by $E_A$. We can also take $L_A = \operatorname{im}(\alpha_1)$, which will be a unital subring. Then take $R_A = E_A \oplus L_A$ as the unital algebra which encodes actions of the operad on $A$.
When regarding $A$ as an $A$-module, we can consider a submodule of it, and we call these ideals. If $I$ is an ideal of $A$, then the module action shows that it is absorbing, and it shouldn't be hard to show that the inclusion of $I \otimes I$ followed by multiplication factors through $I$. We should also verify that $A/I$ is a well-defined $O$-algebra in $C$.
Say that $A$ is simple when the underlying object of $C$ is rigid and simple as an object of $C$ (i.e., has a dual object and the only subobjects are 0 and $A$) and $E_A \neq 0$. One should verify that this implies that those subobjects underlie the only two ideals of $A$. Then say that $A$ is semisimple when it is a finite direct sum of simple $O$-algebras.
I haven't checked all the details, but this passed my sniff-test, and should recover the given definition for $C = \operatorname{Vect}_k$. I'm somewhat dissatisfied, since it's a somewhat external viewpoint, rather than saying semisimple algebras are Artinian semiprimitive algebras, for appropriate definitions, and then proving that this is a sum of simples. If I had time to be careful, you could probably dispense with the assumption that $C$ is abelian. We just (seem to have) needed that it has images, subobjects, (quotients,) a terminal object, duals for some objects, and finite coproducts which interact well with $\otimes$. I'm also not sure whether rigidity is the correct replacement for finite-dimensional, perhaps writing the underlying object $A$ as a finite directed colimit of suitably small objects works, or maybe it's enough to just be simple as an object in $C$.
