Re{ } component of a phase defined function
The definition is:

$$ e^{j\phi}=\cos(\phi)+j\sin(\phi)$$

and your solution is correct.

What exactly do you mean by


  What would the Im{} of the same function be? 


Changing Re() to Im() inside the function gives
$$2Im(e^{j\omega t}+2e^{-2j\omega t})-2=\{2\sin(\omega t) + 4\sin(-2\omega t)\}\rlap{\backslash}{j}-2$$

On the other side, the original f(t) is real, so Im(f(t))=0



EDIT: Had an extra j in the last equation. The result is real and does not contain any j.