Her name is not Elizabeth, it's actually just "Riza". Elizabeth is just her codename, used a few times in the course of the manga. And though her exact age is not revealed, it's a consensus she's between 24 and 29 years of age. In the manga, however, there're hints that she's actually at least 2 years younger than Roy (being in Military Academy at the time of Ishval's anihilation) and not more than 3, 4 years younger than him, since Arakawa more than once has brought up the subject of "Riza not being that young, anymore". My guess is that she's about 27, 28 years old.