Personally - only two people, close friends, know I'm in T, and they only found out this year. I've been seeing this T for almost 4 years. For me, T is private, and it's no one else's business. I don't think that telling people you are in therapy is necessary for progress. Along those lines, any "homework" my T gives me is also intensely personal and private, and not something I'm going to want to share with others. It would bother me if someone else, especially someone I don't know or trust saw me working on it.

As for not doing what your T asks, well, only you can determine that. Are you taking in what your T tells you? Are you just not able to do these things yet, but actually taking it in and thinking about it? Are you actively resistant? If so, maybe that's something you need to explore with your T. Maybe it's time to talk about why she feels you're not doing things she suggests, and why you feel like you can't or won't do them. If you're not a danger to yourself or others, then T's can't force you to do anything. It's up to you to take in their advice and decide if it is something you feel like you can do and that would be helpful for you.