Psychologist-in-training here. My general advice is to talk to your therapist about all of this. Whatever reaction she has, therapy is a place where the two of you can process those reactions together. 1. I doubt your therapist will be embarrassed. She will probably be glad you felt comfortable enough to share it with her.
2. It's completely understandable that therapy relationships may begin to feel similar to friendships. Therapy relationships are a unique kind of relationship, though. Psychologists are bound by an ethic's code. Taking a walk around campus with your therapist would be a violation of your confidentiality (privacy) - others on campus likely know she is a therapist. Some psychologists will engage in the behaviors you mentioned (taking walks, giving gifts, making home visits), but in my experience they are typically older, well-respected psychologists at the end of their career (i.e., Yalom) or they are working in a program where making home visits is routinely done for all clients. Psychologists can get in trouble for doing some of these things depending on the circumstances. 3. The research lab I work in studies child maltreatment, including abuse and neglect. My advisor told me that she once had a student work in her lab who, several months into it, disclosed her own history of abuse and stated that working in the lab was just too much for her. My advisor understood, and the student stopped working for her. I wanted to share this with you just to let you know that you may or may not be happy working in this area (it sounds like you haven't started this line of work yet), and that's ok. As you can tell, my main concern about this plan isn't about who you might run into now and then at a conference.

Please discuss all of this with your therapist. My goal in responding was to give you enough information to feel brave enough to share these things with her. Nothing anyone posts on this forum should ever take the place of discussing these issues with your therapist.