What is the expressive power of finite-state transducers and pushdown transducers? One would assume that a finite state transducer can perform any translation such that the resulting string is from a regular grammar. Similarly, that a pushdown transducer can generate strings from CFG grammars.

I am, however, unsure about this because we can construct a finite state transducer which translates a string of the form $ab$ to $\{ a^n b^n : n \ge 1 \}$, which is irregular.

Is this the case? If so, can anyone provide the relevant literature?
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.