From this answer I understand that you can't actually catch her. However, at the end of the chase I think I was awarded experience. Is there anything you can do during the chase scene to increase the experience gained by this encounter and can your actions at any point during this encounter change the way the cut scene unfolds? Also, what happens if the Cerberus agent gets too close (out of instinct I just shot her until she wasn't coming for me)?
No you can't change the way it unfolds. If you get too close when following then she will attack you and if she gets too close in the final scene she will kill you.