I think she just likes you and perhaps views this too much as a friendship. You've said how much she talks about her daughter to you, and that's the sort of thing friends would do. People share the love and pride they feel in their kids with people they like and care about. She WANTS to share that personal stuff with you, which is not professional nor is it practicing good boundaries. She doesn't have to be a blank slate, I see nothing wrong with you knowing her daughter exists, but talking about her frequently and wanting to introduce you to her is just so beyond healthy that it's worrisome.

So I think the 'need' you fulfill in her is a need on her part for a friendly confidant or girlfriend of her own. Perhaps she doesn't have that in her personal life? Perhaps she sees all the qualities of a good friend in you and she is using your therapeutic relationship to give her a taste of friendship. Your therapy should be about you. It shouldn't be about her daughter or her love for her daughter or any other personal stuff on her part. The texting and emailing in between isn't necessarily bad, but you've said it's rather frequent. Is it therapy related or just casual chit chat?

My concern is that you will get hurt. My concern is that she will string you along in this
'therapeutic relationship' as long as she can to keep you as a 'friend' and to continue to enjoy her time with you and enjoy your personality which she clearly likes. My concern is that you may not get the healing you deserve and that you may start to censor yourself for fear of turning her off or losing her goodwill and high opinion of you. My concern is that when it ends you may lose both your therapist and a friend. And that will be so painful. It's hard enough when therapy ends, but when it's been both therapy AND friendship rolled into one, it's ten times harder.

She may be a great person and a great therapist, but she may not be the best therapist for YOU. YOU may be someone she just clicks with too much and likes too much and therefore actual therapy isn't the best fit.

But you definitely should not feel guilty for that session. You did nothing wrong and you were even kind enough to urge her, several times, to just cancel. (something that was kind, yes, but technically something you should not have worried about as you are paying for that time and you are the client (boss) and it's about YOU, not her, which is another thing to kind of worry about...your concern for her).

She messed up that time for sure. Whether or not you think she's messing up in other ways I've mentioned is for you to ponder. I know this is your first therapy adventure, so you don't know really what's right or wrong or normal or not normal, and there are huge variations anyway, but just keep in mind that therapy is supposed to end someday. Perhaps discuss with her how she pictures that. Ask NOW about whether or not she keeps in touch with former clients and how often that is and in what way. I don't know, I just think right now it feels special and loving but I worry about later on.