There seem to be several questions here. Let me know if I'm on the right path. All of the above seem symptomatic of micromanagement and inability to delegate. However there may be some valid reasons for the behavior, as she might have her side of the story and problems she is trying to solve. A few strategies you could try to make things better:
1. The employee who was in the position you are backfilling did not
properly transition the work to you, but instead continues to do
that work.
Ask to be trained and for proper transition of work assignments. This should involve supporting documentation and related resources to help you get oriented to the work.
Also, Ask for clarity about your job duties and clarification how your tasks differ from the tasks this person who got promoted does. If there are tasks she assigned to you but ended up doing herself, say you would like to understand the reasons and ask how you can do them better so that they do not require rework.
Frame this as you trying to be helpful and make her life easier, rather than create extra work for her, so she can focus on things that are more directly her responsibilities (like project management or whatever). If everything you do is a team project and it is considered normal for several people to 'touch' the work before it is done, get her to confirm that that's part of normal operation and how things work, and stop worrying about it.
2. You are not receiving adequate training on how to do the work in your new position.
Ask her to suggest specific ways to get trained in order to improve performance. What do you need to learn about that you don't currently know? Who can teach you? Where should you go for help? What materials are available to help guide your learning and work?
3. She blocks you from having access to management and 'filters' your messages in her own way.
Being blocked from having access to management and having your words 'reframed' may not be an issue, but this person's way to protect both her boss, herself, and you from miscommunication. She might know how to deliver and frame messages to upper management better than you, and she probably feels responsible for ensuring that the message is properly understood. As someone in a higher position, she may feel on point to ensure appropriate messaging, and she might be right. I recommend that instead of making this into an issue, you let it go and let her do her thing. Let her be the 'front' of the team if she feels that she needs to.
Over time as you gain experience and gain her trust, she will begin to let go herself and you will have more opportunities to message to the management directly. For now, I suggest accepting things as they are rather than trying to fight it, as it could only cause more problems for you. I know it can be hard, but try to be patient and give it time.
4. She complains that you are not keeping her updated enough while you feel that you provide sufficient status through email and other documents.
Ask her for specific examples where she felt she was blindsided, and show her where and how you did in fact provide these updates. Ask why that was not sufficient, and how you can improve your communication so that she feels adequately informed. Instead of arguing, have a dialogue, ask for feedback. Would a weekly half-hour meeting help her to get up to speed on your progress, before she has to report it to management? Would bullet-point summaries of your weekly progress help?
Have her suggest ways that you could keep her better informed, then follow her suggestions. If she wants more frequent status updates, give her that. As long as you are giving her exactly what she requested, she will not be able to keep complaining as the ball will be in her court, so to say.
Generally speaking, you are in a new role and she has been in those shoes. She might be making your life easier in some ways you don't realize, so try to make the most of her in that capacity as a more senior colleague and a resource, not an obstacle. She's just trying to keep things running smoothly and if that means you get stuck in the backseat for some time until you gain her trust, so be it. It's not the end of the world. But hopefully the thoughts above can help move things in the right direction.
Good luck!
