What is the expressive power of finite-state transducers and pushdown transducers?
we can construct a finite state transducer [that maps $\{ab\}$ to $\{a^nb^n n \geq 1\}$].


No, we can't. REG is closed against finite-state transduction.

Wikipedia hints as much without citation; planetmath.org lists some standard references, in particular Hopcroft/Ullman.