Representations of $D(G)$ as an object in the center of $\operatorname{Rep}(G)$

1)2) is standard for an arbitrary f.d. Hopf algebra $H$, as you say it's not hard to idenfity $D(H)$-modules with Yetter-Drinfeld modules. Then, given two of those, say $V,W$ you can define a braiding by $$V \otimes W \rightarrow H \otimes V \otimes W \rightarrow H \otimes W \otimes V \rightarrow W \otimes V$$ where the first map is the coaction of $V$, the middle map the flip, and the last one the action on $W$.

3) it really depends on what you mean by unknown, there are lots of things that can be said. For example, if $C$ is an arbitrary finite tensor category over $\mathbb C$ say, then $Z(C)$ is semi-simple iff $C$ is, and the global dimension (the sum of squares of dimensions of simples) in $Z(C)$ is the square of the global dimension of $C$.