I can understand your feeling uncomfortable. I would probably be a bit weirded out by my T doing this (though I'd probably like it at the same time), but it's partly because I've been seeing him a long time and know that it would be unlike him to do something like that. I mean, I was slightly weirded out when he emailed me an article out of the blue a year or so ago because it made him think of me. I did like it, but wasn't sure what to make of it.

It could be this is just how your T is with clients, that she's just a bit more familiar and friendly. It may just be her style. If it makes you uncomfortable, you could try saying something about it, even if it would be awkward. Things like calling you "hun" may just be part of how she always talks, so those could be more difficult for her to change (though she could certainly try if it really bothered you). With the selfie, I'd say maybe let this one go, but if she does it again or is sending you random texts that don't seem related to therapy, then say something. (I don't mean things like checking in to make sure you're doing OK or appointment reminders, but more seemingly random stuff.)

I don't think it's that odd that you know her birthday, certainly not odd that you know her age. With our former marriage counselor, at one point we just asked his age, and he told us. My current T was a bit more cryptic about his age at first, but then became more open about it, like, "Now that I'm over 50...." (I know when his birthday is, but not because he told me.) Ex-T never told me her age, even when I asked, though she did give me a hint to the age range. So I think it's just a matter of style, too.