Is there a description of finite groups whose all quotients have trivial center? Is it true that only direct products of non-abelian simple groups have this property?



Finite groups with centerless quotients
The answer to your second question is negative. Take a finite simple group $G$ and its nontrivial irreducible representation $V$ over finite field. Then the semi-direct product of $G$ and $V$ has a unique nontrivial quotient, the group $G$ itself.


