The OP asks more than one different things: the classification of fin dim, semisimple, braided Hopf algebras is still an open area (up to my knowledge of course).  

The classification of low dimensional hopf superalgebras (the super- here is to be understood as the simplest case of braided) is another thing. It is still open in general, but a particular case of it has been solved long ago; this has to do with all the finite dimensional (super-)cocommutative hopf superalgebras. These have been classified by means of the following: 

>Let $\mathcal{H}$ be a finite dimensional, super-cocommutative, hopf superalgebra over $\mathbb{C}$. Then $$\mathcal{H}=\mathbb{C}[G(\mathcal{H})]\ltimes\bigwedge V$$
where $V$ is the space of primitive elements of $\mathcal{H}$ (regarded as an odd vector space), $\bigwedge V$ is its exterior algebra and $\mathbb{C}[G(\mathcal{H})]$ is the group algebra of the the finite group $G(\mathcal{H})$ of the grouplikes of $\mathcal{H}$. In other words, $\mathcal{H}$ is a supergroup algebra. 

The above proposition, can be seen as a corollary of the super-version of the Cartier-Constant-Milnor-Moore classification theorem.  
(For the super-version of the theorem, see: theorem 3.3, p.224, [B.Kostant, "*Graded manifolds, graded Lie theory and prequantization*", Differential Geometrical Methods in Mathematical Physics. Lecture Notes in Mathematics, vol 570, p. 177-306, (1977)][1])


  [1]: https://link.springer.com/chapter/10.1007/BFb0087788