I don't know about this one.
One part of me is thinking: Be aware - he is cutting a few boundary lines very fine. Some (definitely not all) new therapists have this huge need of needing their clients to like them and they can be a bit blurry when it comes to achieving that. Some are flirty, others practice a lot of self-disclosure, others are trying to flatter their clients in ways that might not be helpful etc.
What strikes me in your post is that you are aware of how this makes you feel. You feel confused, a bit uncomfortable, it sounds as though you feel torn a little... That usually is a recipe for issues that are beyond the issues you are trying to address in your sessions.
Even the buying a Starbucks gift card is a red flag for me. I think gifts are difficult in therapy.

But on the other side, if you feel he can help you with your issues, if you feel you can make progress, then the flattery and the slightly flirty advances of your T might be a helpful way to build a healthy attachment.

But that said, my stomach feels funny reading about this so please be good to yourself and trust your instincts.

Amelia