The center is trivial : as Benjamin said in the comments, an endomorphism of the identity is the identity on $\Delta^0$, and using the maps $\Delta^0\to \Delta^n$ you find that any endomorphism of the identity is the identity.
For the trace, you can use the following fact coming from unwinding the definition : if $f,g$ are composable in both orders in $C$, then $tr(fg) = tr(gf) \in Tr(C)$ where I let $tr$ of an endomorphism be the class it defines in $Tr(C)$.
Using this, for any endomorphism $f\in \Delta$, writing it as a surjection followed by an injection and inducting on size shows that every $tr(f)$ is $tr(id_S)$ for some $S\in \Delta$ (note that any automorphism is the identity). Furthermore, $f\mapsto $ the cardinality of the fixed point set of $f$ defines a morphism $Tr(\Delta)\to \mathbb N$ (this is not obvious, but a fun exercise in combinatorics) which distinguishes all the $tr(id_S), S\in \Delta$ so that $Tr(\Delta)\cong \mathbb N_{>0}$