As a teacher, "Danny is wearing trainers instead of school shoes because he stepped in a puddle on the way to school" is fine for notes, but when the issues are so important like this, I want to sit down face to face and nut out the issue. It's quicker than waiting for notes to go back and forth, and has the benefit of personal communication. We can also usually involve the student in parts of the meeting. I then send a summary out to everyone concerned and we confirm what was said and agreed to.
OP, sit down with the teacher and have a chat and discuss some strategies. It always works better if you go in with solutions, rather than just complaints. If she's not getting time to write down her homework, then she'll need extra time for that, or alternative strategies put in place. I usually put the homework up on a smart board with a list of what is needed. Students can then copy it down, and if they have to stay back a few minutes to make sure they have it all, then so be it. Perhaps put a laminated page in her bag which lists all the things she should bring home on the average day (reading book, coat, pe kit, etc) that she can look at each afternoon before she leaves, with the teacher checking for any usual items (special homework, etc) each day? If she's forgetting to bring her books or homework TO school, then that's not something the teacher can control, you need to take some responsibility for that. You'll need to check every morning to make sure she's got what she needs before she's out the door. The checklist system can work well there, or have her sit down and do her homework, then repack her bag the night before, not when it's crazy and busy in the morning. Homework done, bag packed, sitting beside the front door ready to go.
If she's getting overwhelmed with the amount of homework, set a time limit for her. I tell parents that if their child has genuinely attempted the homework for x amount of time (age appropriate) that they should stop and the parent should let me know why so that I can work on what it is that they are struggling with. I don't want any child sitting at home getting upset over some homework tasks (not that I do much in the way of formal homework anyway).
Good luck with your chat with the teacher, I'm sure they want to help your daughter as much as you do but it is a team effort.