What is it called when every object of a category is the quotient object of some free object? For example, every group is the quotient of a free group. Is there a name for this property in a general category, or does this property necessarily follow from the universal property of free objects?
 A: It depends on what you mean by "quotient object," but broadly speaking, this says that the free object on one element is some kind of generator of the category, for various values of "generator" (see this blog post for a lengthy discussion of what "generator" can mean). 
First, note that to define "free object" you need more than just the structure of a category. What you need is really a concrete category: a category equipped with a forgetful functor $U : C \to \text{Set}$. You can then define free objects, if they exist, using a left adjoint $F : \text{Set} \to C$ of this functor (if it exists). 
One thing you might mean by "quotient" is "epimorphism." So you might ask that every object admits an epimorphism from a free object. If $C$ has all coproducts, this is equivalent to the condition that the free object $F(1)$ on one element is a generator in the usual sense, namely that $\text{Hom}(F(1), -) \cong U(-)$ is faithful (which in our setup is always true, since we assumed that $U$ is faithful). 
But epimorphisms don't always behave like quotients. For example, in commutative rings, $\mathbb{Z} \to \mathbb{Q}$ is an epimorphism, but it seems strange to think of $\mathbb{Q}$ as a quotient of $\mathbb{Z}$. A better way to say "quotient" is regular epimorphism, and this corresponds to $F(1)$ being a generator in a stronger sense. 
