That's awful. As everyone else has said texting and just looking at her phone is very unprofessional. I think note taking is quite distracting and my understanding is that your fee should cover the time it takes T to make notes in your chart after you leave. The exception would be the first few sessions where they might be getting basic info about you (eg writing down names of the important people in your life etc).

If she must take notes and must use her iPad for it, it should be on airplane mode so she isn't getting texts or calendar pop ups at all.

It also sounds like she is uncomfortable with silence and passing that awkwardness on to you with "so what else do we need to talk about?" I don't agree that it's her job to keep things running smoothly. I think the expectation that things need to run smoothly can generate a lot of anxiety for you and her job is to let you know that she's okay with however things are running, she can cope with silences and she can cope with neither of you knowing what comes next.

I wouldn't see the point in going if I didn't feel I had T's undivided attention.