The reason Kai can't find Erica in 2019 is because she's a therapist at that point.Thus she no longer exists in the 'real' world in the sense of having an identity that he can actually find.
Doctor Tom is The Doctor.And I didn't even say "X is a time lord" ... ("facepalm") Just think about it. He is a DOCTOR! He travels through TIME AND SPACE! His office is BIGGER ON THE INSIDE! (my theory is that his office IS the doorways that Erica walks through)
Erica's voice overs are things she's saying to other people when she's a therapist.The voice overs seem to speak in an older more experienced fashion and almost in a more teaching fashion as if she's telling people things.