A lot of it sounds like a mother unwilling to share her child with her father. It is not YOU. It is your PLACE. If she wants on some level to be with him (or at least have him not be with anybody else) and you are there in the way, then she will just be angry that he has someone. My DH's ex used to sign things The Mom, like anybody would forget who she was.
Your DH needs to do things like navigate communication with his ex. This is not your direct job. Let him arrange things with her, fill her in on doctor visits, etc. When possible, keep it to email. Be professional. Don't get into arguments. Follow the court order. Etc. He needs to buffer you (and any children you may have, and his DD) from his ex's problems. It is his call if he wants to have more communication and to keep it to the kid. It's a professional relationship now.
You are probably doing nothing wrong. My DH had custody when his kids were little. By default, I did a lot for them, more than their own mother. I took them to school, I picked them up when they were sick, I packed their lunch, did homework with them, etc. Not because I was replacing anybody but because someone needed to do it and I was there. It didn't always make their mom happy, but I couldn't (example) let the kid go to school in shoes that didn't fit. My DH had to get the kids 2+ hours round trip in the summers and often I went (or I was sometimes the person driving them). I tried to be low key - when they were old enough they just hopped out of the car and went in. I didn't carry their bags or go to the door. Many many times their SF also transported them, so their mom couldn't say much if she had her DH doing the ferrying, too.
It is the fine line many SMs walk - love the child, but heaven forbid you seem to love her like your own (as many would want you to do) because the mother will get upset.
So find your own path. Do what needs doing when it needs doing. If your DH wants company or it is silly to leave you behind, then go along. Let her mom handle things like the first bra if she's an involved parent. But don't cut yourself off from supporting your DH or push yourself in the background all the time to keep the peace. If your DH is capable of holding the mother to boundaries and keeping things about the KID, then let him.
You ARE the stepmom. I think the more of us that are sane, reasonable, caring people who proudly wear that title the less the fairytale will be the first thought. That's your title, and there's nothing wrong with it. There will be times when a well-meaning stranger calls you mom and you just roll with it, and there will be times when it is appropriate to correct them. You'll know when you get there.
I never had a heart to heart sit down with my DH's ex. There was never a need. I just gave her respect (more than she often gave me) and tried to keep my eye on the long view. Both sks are grown now and the interactions with their mother are minimal. They organize their own lives. It takes a loooong time for a family to settle so don't get caught up in today. Keep the big picture. It is a transition for everybody.
Once, when I was venting about BM, my SIL calmly looked at me and said, "We've had 10 years to get used to her behavior. You'll get there." Which is true. There is a lot less angst on my part once I learned to let go of little things that I made too important. Also, remember that her relationship with your DH has no bearing on whether or not she's a good mom. My DH's ex is not all bad. She has done some good things for her kids and loves them in her own way. Her way does not need to be my way or DH's way. If this mother is a good mom, then her communication woes with your DH are secondary to the well-being of the child.