I'm so sorry you had a difficult session, but as the previous replies mentioned (wonderful ideas, btw), your T probably told you about her daughter because she wanted you to know why she has to end on time every week and it's not because she does not want you in her office another minute. However, I totally get your pain at being sidelined by her kid as I get super jealous of my T's son too (and he's only about 7 or 8) yet I cannot expect her to care for me more than her son.

In my sessions I sit facing the clock and when I think I tend to stare at it a lot. I try not to bring up hard stuff towards the end of a session out of courtesy to her but once we were running over time and she asked me if I had suicidal thoughts. I guess that was too imperative to put on hold and she took another 5-10 minutes (despite having another client waiting) to make me contract for safety.

I think your T should have had a bit more sense than to purposely induce such hard topics. As for the phone-checking thing, it's at least comforting to know that she doesn't actively check her messages, but if you're really unhappy about it perhaps you could mention this in your next session? Mine keeps her phone face down on her desk (away from where we sit) but she gets a LOT of notifications and her phone vibrates loudly against the table.