The crux of what I wish to know is what results from representation theory, a subject usually framed within the category $\text{Vect}\_\mathbb{k}$, follow in more general braided monoidal categories? I would be very satisfied with references to texts that cover this.

I'll try to be more specific. Let $\mathcal{C}$ be monoidal, abelian, complete under arbitrary countable biproducts, and enriched over $\text{Vect}\_\mathbb{C}$. An algebra in $\mathcal{C}$ is an object $A$ with morphisms $m:A\otimes A\rightarrow A, u:1\rightarrow A$ satisfying suitable axioms, and a left $A$-module in $\mathcal{C}$ is an object $V$ with morphism $a\_V:A\otimes V\rightarrow V$, again satifying some conditions. If $\mathcal{C}$ has a braiding $\psi$, we can define commutative algebras as those algebras such that $m\psi=m$.

For instance take $\mathcal{C}$ to be $H\text{-Mod}$, the category of (finite-dimensional) modules over quasitriangular Hopf algebra $H$ (the quasitriangular structure on $H$ makes $H\text{-Mod}$ braided monoidal). Recall the following classical (i.e. in $\text{Vect}\_\mathbb{C}$) result from representation theory: for commutative algebra $A$, every simple finite-dim'l $A$-module is $1$-dim'l. Is there an analogue of this statement for braided commutative algebras in $H\text{-Mod}$?