Is there something resembling a characterization of which groups can map onto a nonabelian free group? Obviously they cannot have property T, and should have nontrivial abelianization, but are there some positive results?

Such groups are often called 'very large'. A group with a very large subgroup of finite index is called 'large'. Here are some miscellaneous facts:
$1\to K\to\Gamma\to Q\to 1$ where $K$ is 3generated and $\Gamma$ is large. So large (and hence very large) groups are quite common. I doubt there is any kind of characterisation, but there are some open questions that are relevant. For instance: Question: Is there a finitely presentable group $\Gamma$ with $vb_1(\Gamma)=\infty$ which is not large? Note that $\mathbb{Z}\wr\mathbb{Z}$ gives a nonfinitely presentable counterexample. (Here $vb_1(\Gamma)$ is of course the maximum of the first Betti number over all subgroups of finite index. It is obvious that $vb_1$ of a large group is infinite.) And another thing... I just remembered that there is a (not implementable) algorithm to determine whether a finitely presented group is very large. The point is that the group $G=\langle x_1,\ldots,x_m\mid r_1,\ldots,r_n\rangle$ maps onto a nonabelian free group if and only if some system of equations and inequations $[x_p,x_q] \neq 1\wedge\bigwedge_j r_j(x_1,\ldots,x_m)=1$ has a solution in $F_2$, for some $p\neq q$. Now, such systems of equations and inequations over a free group $F_n$ can be solved by Makanin's algorithm. 

