Yes, what you say makes sense. It feels a little less... boundaried or showing less professionalism. More like a friendship (personal disclosure, sending selfies..), especially so soon in your work together.

I don't think she means anything 'bad' by calling you hun and her way of showing affection but yes, it can be uncomfortable. Not making excuses for your T but maybe that is the 'normal' for her generation.

Yes, one should be able to discuss anything with T. Just a word of caution: not all Ts are able to respond without getting defensive or reacting 'badly'. So, I am not saying don't bring it up if you wanted to but maybe be attuned to T as to *how* you bring it up / her reactions (so you don't shut down, freeze up, end up being hurt etc.)