does every CFL have an ambiguous CFG?
Every nonempty context-free language has an ambiguous grammar. Consider any context-free grammar for the language with starting symbol $S$. We add new non-terminals $S',A',B'$, make $S'$ the new starting symbol, and add the following rules:
$$
\begin{align*}
&amp;S' \to A' \\
&amp;S' \to B' \\
&amp;A' \to S \\
&amp;B' \to S
\end{align*}
$$