Is there non-trivial examples for groups $G$ satisfying $G = \mathrm{Aut}(G)$

And for an example where $Z(G)\neq \{e\}$, so that the automorphism group is not just the inner automorphism group, take the Dihedral group of order $8$.

If $G$ is a complete group, that is, with trivial center and no outer automorphisms, then $$G\simeq \text{Aut}(G)$$ as every automorphism is the conjugation by some element, and the map $g\mapsto g\cdot g^{-1}$ has a trivial kernel.

Yes, for example we have $\text{Aut}(S_n) \cong S_n$ if $n \neq 2,6$.